Question
The cartesian equation of
is
Hint:
The term "polar coordinate system" refers to a two-dimensional coordinate system where each point's location on a plane is determined by its distance from a reference point and its angle with respect to a reference direction. Here we have to find the polar equation of
.
The correct answer is: ![open parentheses x to the power of 2 end exponent plus y to the power of 2 end exponent close parentheses to the power of 3 end exponent equals 4 a to the power of 2 end exponent x to the power of 2 end exponent y to the power of 2 end exponent](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAJgAAAAWCAYAAAA8eFWJAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAATRJrTQAAAAyxJREFUeNrtWT1oFEEUHq4IIiKIHCIWgSCSQo5AKhGxERGRI11IISKBIBIkvRwiQTgsUkgaCwuRIEiwEAmBECQcIQhBQhALwcLCIggWIhKOwPnGfBeHdWZ3bubtzazuBx/c3e7NvHk/M++9EaJEB2wTt4iXS5WUyAvDxM+lGkrkhQrxW6mGuI+ZdeLpAspxnDhPvFnaLl77yx3gNvFdBDtRL3J0FXT9Pw5oDttljiEfNhmE3Y3EEKocTazPhKPEB8Spgh3p3AG9m9cYNeIGw+AXiWsRKF8nxzrWWYTgCBHQHLYzjiGVP+I5+EmMMxRY4SY5RomtlP9dIG72Qb5rOOJERAHNYTvjGCN44IMzxNeYJCSy5GjB0ZL5jNwFlomDOct3hPiBycG4AprDdqljPCROey50CXmMS8LKuXNlyXGHKc90xWPiJMO6uQLax3bWYyzjeEhi0mCMWTzrQg4+7FER2eAJcVzz+wySc1s5zhFXAznXeWXuTkrifp+4I/7cLgw6OAWX7Vj0/p04YHgmK5QTynfZJ5o3lM0quR3sBnZaFYeJH8V+D8tWjgGs16c9kEYT5LzbirOY3pW70iPiMTjbC+JY4h3bgOawHYve91KEvEKcw+dLOUS/rYNdhbJV3CM2HOZsB9i9mok0RLfuCc0an2LtLgHNYTsWve9lPF9BtbKleK3o8y4g532vfK8SPxEPFcDBapoiSrfONzjCVax5Fh6+tmPRe9oR2T1v2xY9pDx3MImfyuc5yNUrQhyRb8Xf3faOYX2VRD72w1O/HLbz1vsKElBTYrqIgScCO1gLJfkQoqjiMF+IJN/WKb9qdO/TpeeynbfeTW0KOeArJHWyf7PJcET6OJjMR+rEBeF+bzitSVpDQLfuL4mjpwEHcQGn7bz1rmu0VlGtqELJDvSzgA42g7J52zMaRyN1sFn0ySrIuxZgg17BbTsOvf++h6wpecpL4inNe89RnYTAGAxTd/x/1lVRaAeTuIuqrYHdbEfTpsjKMblt56v3g0pnQ8SNuqeMNpfdsaH6D+j9ALdE2GuULCxpyvhe8swpUaLfei8Mzor9LneJAun9F8VbFxGLOYnjAAABSHRFWHRNYXRoTUwAPG1hdGggeG1sbnM9Imh0dHA6Ly93d3cudzMub3JnLzE5OTgvTWF0aC9NYXRoTUwiPjxtc3VwPjxtZmVuY2VkIHNlcGFyYXRvcnM9InwiPjxtcm93Pjxtc3VwPjxtaT54PC9taT48bW4+MjwvbW4+PC9tc3VwPjxtbz4rPC9tbz48bXN1cD48bWk+eTwvbWk+PG1uPjI8L21uPjwvbXN1cD48L21yb3c+PC9tZmVuY2VkPjxtbj4zPC9tbj48L21zdXA+PG1vPj08L21vPjxtbj40PC9tbj48bXN1cD48bWk+YTwvbWk+PG1uPjI8L21uPjwvbXN1cD48bXN1cD48bWk+eDwvbWk+PG1uPjI8L21uPjwvbXN1cD48bXN1cD48bWk+eTwvbWk+PG1uPjI8L21uPjwvbXN1cD48L21hdGg+/7ZHMQAAAABJRU5ErkJggg==)
Using the formula, we may generate an endless number of polar coordinates for a single coordinate point.
(r, +2n) or (-r, +(2n+1)) are possible formulas, where n is an integer.
If measured in the opposite direction, the value of is positive.
When calculated counterclockwise, the value of is negative.
If laid off at the terminal side of, the value of r is positive.
If r is terminated at the prolongation through the origin from the terminal side of, it has a negative value.
The equation is given as
![r equals a sin space 2 theta
N o w space w e space k n o w space t h a t colon
sin space 2 theta equals 2 s i n theta cos theta
A p p l y i n g space t h i s comma space w e space g e t colon
r equals 2 a s i n theta cos theta
M u l t i p l y i n g space b y space r squared space o n space b o t h space s i d e s comma space w e space g e t colon
r cubed equals 2 a left parenthesis r s i n theta right parenthesis left parenthesis r cos theta right parenthesis
P u t t i n g space r cos theta equals x space a n d space space r s i n theta equals y comma space w e space g e t colon
r equals square root of x squared plus y squared end root
left parenthesis x squared plus y squared right parenthesis to the power of 3 divided by 2 end exponent equals 2 a y x space
S q u a r i n g space b o t h space s i d e s comma space w e space g e t colon
4 a squared x squared y squared equals left parenthesis x squared plus y squared right parenthesis cubed](data:image/png;base64,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)
Here we used the concept of the polar coordinate system and also the trigonometric ratios to find the solution. So the equation is .
Related Questions to study
Two tuning forks
and
are vibrated together. The number of beats produced are represented by the straight line
in the following graph. After loading
with wax again these are vibrated together and the beats produced are represented by the line
If the frequency of
is
the frequency of
will be
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAPcAAAC7CAYAAAC5FghEAAAgAElEQVR4nO3deVSV95348Tf3XvZ93xEQUVF2BBURN9yiMTVN2mpN0sS002Y66ZxpT2emnV/bM+m0087MmdOettPW2iYaW1MbE5cQq6IiooBhE2TfleXChQsXLtz1+f3h4Z5QWZWd7+scz4nc53nuB3M/9/s83+XztZIkSUIQhEVHNtcBCIIwM0RyC8IiJZJbEBapRZncDx8+RKvVznUYgjCnFlVym81mNBoN//7v/05paelchyMIc0ox1wFMp/7+fi5dusS1a9fw9fXF09OTFStWzHVYgjAnFk3LbTabUSqVnDhxgtbWVjIzMykuLp7rsARhziya5O7r6+PevXtkZWWh1Wq5d+8en3zyCR0dHXMdmiDMiUWR3JIk0djYyLlz55DL5QDIZDLy8/O5cuXKHEcnCHNjUSS3TqejsrKSmzdvcuTIEVxdXdmyZQsmk4mcnJy5Dk8Q5sSiSO6ysjJqamr4zGc+w7e//W08PT05ePAgr7zyCg4ODvzlL3+Z6xAFYdYt+N5ynU6HWq3G19eX559/nsDAQBQKBR4eHqSkpODp6UlbW9tchykIs27BJ7fJZCIoKAg/Pz+ioqIwm82W1wICAkhOTqaxsZHe3l6cnZ2RyRbFzYogTGjBJ7dcLmf58uVYW1uP+rqfnx9eXl709PQgFsAJS8mCT25bW9sJj1EoFHh7e89CNI+TJAmDwUBjYyNyuRwvLy9cXV0nPEev12NtbS3uNIQnJj45M2w4sffs2cNnP/tZMjMzJzzHbDZTXV2NRqMRdxvCExPJPcOUSiVvvfUWy5YtQ6lU0tLSMuE5Wq2W73znO2RkZPDb3/4WtVo9C5EKi41I7hnU1dVFdnY2ly9f5sUXXyQ4OJienh56e3vHPEej0ZCXl0dRURFKpRJra2scHBxmMWphsRDJPYMaGhrIzMwkMTGR9PR0vLy80Gg047bEPT09XLp0CbVaTUREBEFBQdjY2Mxi1MJiIZJ7hiiVSgoLC2lsbOSNN94gJCQEFxcXBgYG6OzsHPUcs9lMV1cX169fR6fTkZycTEhIyCxHLiwWIrlnyCeffMKtW7eIiIhgz549ODo64u/vj8Fg4MGDB6Oeo9VqefDgAVVVVTg5OZGcnExQUNAsRy4sFiK5Z4Beryc3N5fKykqeeeYZent76e3tJTg4GHt7e3p6ekY9r7Ozk9LSUnQ6HampqURERODo6DjL0QuLxYIf556P3n//fW7cuEFJSQlf+cpXsLOzA2BgYIDg4GBWrlw56nmNjY1kZ2cjl8uJjY3F2dl5NsMWFhmR3NOstraW8+fP4+fnx69//WuCg4Mtr5WVlXHhwgVqa2sfO0+j0VBRUUFhYSFubm7s2rULLy+v2QxdWGREck+zjz/+mM7OTrZt28aBAwdwd3e3vObh4cHt27dRqVSo1Wrc3Nwsr9XV1VFSUoJOpyM2Npbly5djb28/F7+CsEiIZ+5pYjAYqK2t5aOPPiIsLIzU1NQRiQ3g6uqKo6Mjg4ODdHV1jXitoqKCe/fu4erqSnp6uljkIjw10XJPE6PRyIMHD3Bzc2P37t3ExcU9doy7uzsxMTG0tbVhZWVl+fng4CDV1dXU19fj5+fH9u3bJzVnfrrp9XpaW1tpbm7GYDBYfm5ra4ufnx+hoaEoFOIjs1CI/1PTxN7eni1btrBly5Yxj3F3d+fNN9987OdNTU1UVVWh0WhITEwkJiZmzFVuU2U2m5EkCZlMNuILZTQqlYq3336b3/zmNxiNRhQKBUajEWdnZ3bu3Mlbb72Fu7v7hNcR5gdx3zcPFBcX09zcTGhoKNu2bcPJyWlaEmi4jnt7ezuDg4MTHm8wGOjs7GTv3r2cO3eO+/fvc+nSJT73uc9x5swZ6uvr0el0Tx2XMDtEcs8Dt2/fpq6ujoCAAJKSkqatZdRoNJw+fZqXXnqJn/zkJ5SVlY17fH9/P/n5+Xh5eVmWpvr4+ODr64vBYMBkMolVaguISO45ZDabKSsro7y8HLPZzOrVq1m7du20XNtkMtHR0cHJkye5c+cOGo3GUhl2LDqdjoaGBsLDw3FycgKgpqaG27dvs3nzZgICAsQ89wVEPHPPIZPJxLVr12hubsbf35+oqCg8PT2n5drd3d3cvn2bwsJCfHx8WLt2LX5+fmMer9VqaWtrQ61Wk52dzcOHD7G3t6eyspLOzk7+8R//EW9v7wm/IIT5QyT3HJEkCZ1OR3Z2Nl1dXWRkZExbqz08LHf27Fm0Wi2bN28mPj7+saG5T1OpVFRXV+Pg4IBSqUSj0QCPviScnJzw8fERib3AzKvkNhqNdHZ2Wj5YVlZWKBQKAgIC5mRoaCYZjUaUSiXl5eUYDAaio6NZs2bNtFxbpVJRUFDAtWvXcHNzY+/evSNmyo1GrVbz4MEDVq1axR/+8Af8/f0BOHv2LD/+8Y/54Q9/yPHjx/Hw8JiWGIWZN6+euZVKJd/61reIiooiKiqK6Ohodu7cSVVV1VyHNu0GBga4e/cuKpWK6OhooqOjx21Zp6KwsNCy+8oXv/hF0tLSJrzd12q1aLVaNmzYMOKLNDw8nNTUVB48eIDJZJqW+ITZMa+S+6c//SnR0dFcv36d4uJiPv74Y+zs7Dh+/DglJSVzHd60UqvVXLhwAa1WS2xs7LSt266vr+fq1avcvXsXf39/vvjFL07qS6Ozs5OKigoCAgJG3H5rtVo0Gg2BgYHitnyBmVe35Z///Ofx9PQkJCQEGxsburq6iI2NpaWlhe7u7rkOb9oMDQ3R1NTE1atXsbKyIjExcdqS++bNm+Tm5uLs7MyuXbtYvXr1hI80/f39PHjwAJVKRVRUlGUCTUNDA1lZWVRXV/Paa6+Jue4LzLxK7pSUFMt/GwwGVCoVzc3NREdH4+LiMqVr/fGPf6SoqAgAJycnUlJSSElJwc7OjtraWvLy8kbc7sfGxpKSkkJQUBBqtZq8vDzy8vIwGo0ABAUFkZKSQmxsLAB3794lPz+f9vZ24NEMteTkZFJSUnBycqKxsZG8vDzKy8st77F27VpSUlKQyWQUFBTQ1taGg4MD/v7+E5Y7noyGhgays7Opra1lzZo1HDx4cFITYjQaDV1dXbS2tvKXv/yF3NxcFAqF5Xc7dOgQ27dvF8NgC8y8Su7BwUGKiopobGxEr9fT1taGTCYjPT19yi1be3u7pQVyc3MjMjLS8sw4ODhIe3s7dXV1luP9/f0ts6+Gv1gaGhrQ6/XAo8694Y4+eLRlcEtLC83NzcCjL5CIiIhx38Pb25vBwUF6enrIycmx1DRXKBRPNWdbkiSMRiNXr16lsLDQ8kWTmJg4qQkxCoWClStXsmvXLvR6PU1NTZbXtm7dytGjR584NmEOSfOIUqmU/uVf/kWKj4+X4uPjpdTUVOk///M/pYKCAqmvr29S1zCZTNKqVauk999/f4ajfTJ6vV7685//LIWGhkpyuVwCpI8++uiprmkwGKT6+npp3759koeHh/Tcc89JWVlZ0xSxsFDNqw41b29v/uM//oPCwkIKCws5c+YMf/7zn/nmN79Jbm7uopj6qFQqqaiosKwgm45lnQMDA5w7d47i4mKsra1JSkoiOTl5GqIVFrJ5ldx/y9XVle9973uoVCoqKyvHrfe9UNTV1VFeXo6bmxv79+9/6prkRqORjo4OTp06hUqlYseOHaSnp4ta58L8eub+WzKZDE9PT0wmE3q9fsQOngtVRUUFRUVFuLi4kJGRYZmk86Q6Ojq4cOEC1dXVBAcHs3PnTmJiYsSyTGF+tNxms5nu7m6OHz9u6eGGR0NGWVlZuLu7ExgYuOBbo6amJsrKylCpVISFhZGWlsaXv/xlli1b9kTXGxoaoqqqijNnzqDRaNi5cycJCQlTHlkQFqd503JLkkRBQQGVlZUUFBQAj3qcb9++za5du4iLi7NUEV2oCgsLKS8vx8HBgdjYWIKDgyecFjqe1tZWcnNzuXfvHkFBQezatUtsYiBYTCm5BwcHkclk2NraWoZfVCoVRqMRJycnnJ2dn2gW0/Dt95YtW7h48SJ37twBwMbGhujoaF555ZUF/6E1mUx88skn1NfX4+/vz4YNG5AkidbWVjw8PKY8QcRgMFBWVkZWVhbW1tY888wzxMbGilZbsJhScldWVuLg4MDKlSsxGo20tbXxwQcfoFarSUhIYMOGDU+1ZPFzn/scn/vc5574/Pmst7fXsnwyJSWFTZs2YTQaeeeddzh48OCYtczH0tnZSUFBAcXFxQQHB/OlL31JlEIWRpjSM/cvf/lLzpw5AzxaCvizn/2MU6dO8d577/G73/2OS5cuzUiQC50kSZSWltLa2kpwcDDr1q3Dx8cHg8HAiRMnRkwamazMzEwyMzNxdnZm3759REVFLfjHFmF6TSm5tVotOp0Og8FAW1sbFy5c4LXXXuO///u/8fPzIzc3d6biXNAkSeLatWu0tbURHh7OmjVrLOPbg4ODU15tVVJSwpUrV6irq2PFihV88YtfxM7OTvSQCyNMKbltbGwYHBykpqaGsrIyHB0d2bBhA5s3b8bNzU1sEj8Kk8lEV1cXN27coLe3l1WrVj31uu2srCzu3buHt7c3mzdvZvny5aLGufCYKT1zx8XFce/ePX7xi1/Q19fHvn378PX1xd7eXtSzHoNWqyU/P5+6ujqcnZ1ZsWKFpdyRTCYjPj5+xM4j4zGZTDQ3N3P9+nW6urpIT09n586dYkGHMKopfd3v3LmTkJAQS/WQF1980dI7GxQUxPLly2ckyIVMo9Fw4cIF+vv7Wbt2LcuXL7cko0Kh4KWXXprUOLckSfT393Px4kVKS0txc3Nj48aNo25+IAgwxZY7ICCAo0eP8sorr2BtbW0pxWMymUhPTxc1rf+G2WxGrVZz8+ZNhoaGSEpKIiIiwvK6QqHgueeem9S19Ho9zc3NnDx5EpVKxcGDB1m/fr3oRBPGNKXkPnbsGH5+fhw6dGjEz1UqFTdv3sRgMFjWOwuPloVWVlZSWVmJo6MjUVFRli/Eqers7OT8+fOUl5fj5OTEpk2bpq2gorA4Tem2/P79+zQ0NGBlZTWiZ9bW1pbGxkaKi4unPcCFTKlUkpeXhyRJbNu2jfDw8BEtrdFo5N1337WsCR/LcDXTP/zhDwwODnL06FHS0tJEZRRhXBO23Gq1mmPHjmEymSgsLKSxsfGxfaxaWlqorq4mMTFxxgJdiFpbW7l+/TpWVlakpqbi5+c34kvRaDRy4sQJli1bNu4MvMbGRq5du8bDhw+JiYlh+/bthISEiB5yYVwTJrfBYKCxsRGj0Uh/fz8ymYyGhoYRx/T19REfH8+OHTtmLNCFRqVScf/+fWpra/Hz8yM5Ofmx2Xtms9myAeBYBgcHLcUi5XI5+/btIyIiQrTawoQmTG4HBwd27dqF2WymtbUVb29v9uzZM+IYW1tbQkNDCQ0NnbFAF5ra2loKCgowGAxs3LiR5cuXP9Gqtvr6enJycqiqqmLZsmXs3btX1A4XJmXC5HZ0dGT//v0AdHV14e3tzbPPPjvjgS1kwy1yYWEhTk5OZGRk4Ojo+NhxVlZWeHp6jlmddGhoiBs3bnDjxg2cnZ3ZvXs30dHRC37pqzA7ptRb/tprryFJ0rjljsQUyEe30g0NDdTV1REUFMSBAwcsG+t9mkKh4MiRI2M+b1dUVHD58mUqKytJTU3llVdeEUNfwqRNeVpZbW0tpaWl9PT0PPba8uXL2bp167QEtpCVlZVRUVGBs7MzmzZtIiAgYNQZfMOTWEZLfIAPP/yQe/fusXLlSg4cOMCKFStEJ5owaVNK7qtXr/LBBx9QW1uLo6MjFRUVREVF0d3djb+//7Rth7PQlZWVUV9fj7e3N+vXr39sdGGYlZXVqP9mRqOR0tJSbt68SVdXFxs2bGD37t1imqkwJVNqBq5evUpbWxtbtmwhNTWViooKNm7cSHJyMvb29ouigOHT6uvro7CwkObmZoKCgli/fv2Yx5pMJnJycujq6hrxc4PBwIULF6itrSU4OJiUlBTRWSlM2ZSSu6mpicDAQF577TV27doFwO7du3n11Vfx9/d/onXJi839+/e5f/8+JpOJiIiIEdNN/9ZwsYZPT2IZrov28ccfMzAwQGpqKhs3bhSttjBlU0puW1tbXFxccHR0RCaT4ejoSE9PD97e3uj1+sfGv5eiGzdu0NraSkhICGvWrBm3A8xkMnH58mU6OjqAR4tDOjo6OHPmDGVlZfj7+7N58+Zp29pXWFqm9My9bNky9Ho9NTU1uLi4EBcXx9WrV2ltbaW7uxtnZ+eZinNBMJlM5Ofno1QqSUxMJCkpaUrnDw0NUVNTw9tvv41Op+PAgQMkJSUtur3JhdkxpZZ7165dGI1Gzp49i4eHB2+88QanTp3iK1/5CvX19eM+Xy52er2eqqoqKisrMRgMhIeHj3tLPpqamho+/PBDOjs7iYmJYe/evYSHh89QxMJiN6WWe82aNZYifI6OjpbpphqNhvDwcKKjo6c/wgVicHDQUm0lJiaG6OjoMYe4hikUCg4ePEhgYCA6nY7y8nIyMzORy+UcOnSIsLAwUQRDeGJT+uQ4Ozvj7OyMXq+nu7sbvV7P9u3bsbKywsHBYUnPnBoYGOCjjz6ir6+PmJgYIiMjJyzz/OliDTU1Ndy+fRuVSkVcXBw7duwQ00yFpzKl5NZoNFRVVVFSUkJLSwt6vZ5vfOMbeHt7U1xcTH9/P5s2bZqpWOctrVZLTU0N+fn52NraEhsbO6k66zKZjNjYWLRaLbm5ueTk5ODs7Mz+/fsJDQ0Vz9rCU5lScldUVHD27Flu375Nb28vJSUlHDlyBB8fH/7617/S3Ny8JJNbqVRy/fp1VCoV69atY/Xq1ZOq3y5JEg8ePODhw4fcvHmT5uZmEhMT2bdvn5hmKjy1KSX3cAIfOnSI5cuXk5GRYXlNo9E8dfVTg8GARqNhaGgIeDSDy8bGBldX13n77ClJEu3t7Vy7dg2z2cz69evx9fWd1LkGg4F33nmH6upq7t69S0hICM8//7yosCJMiyn1lldVVeHn58fzzz8/6Q/wVFRXV/P3f//3REVFERUVRXx8PK+++uq8Hj/X6/W0trZSWlpqqbgSFBQ0qXOHJ7H86U9/oqWlhejoaHbv3j3DEQtLxZSS22w2W1rTTy9g0Gq1ltb2aRw/fpw1a9bw3nvvce7cOX74wx+Sl5fH6dOnqaure+rrz4Tm5maKioowGAxs3ryZFStWjLq8cyx6vR69Xk9MTAzbtm174hprgvC3pnSvGxUVRUtLCx9//LHlmbK7u5v6+no0Gs1TlzZ+9tln8fHxITw8HIVCQXBwMElJSdy/f5+2trZ5WTq5traWvLw8bG1t2bp1K+7u7pNauaXVaikuLkav12Nra8v69evZtGnTmItMBGGqppTcW7duJTMzkw8++AC5XI4kSfz+979Hr9fj6+tLWlraUwWTnp4+4u8uLi74+voyMDAw5S13ZsPg4CDV1dWUl5fj5ubG1q1bJxzbhkcz2drb23nvvffo7+9n9erVrFu3bsHvZCrML1NK7vj4eAYGBujp6aG8vJxNmzZRV1dHeHg4qamprFu3btoCM5lM9PX1UVVVRXx8/JS3pi0oKLC0oHZ2doSFhREWFoa1tTVKpZKGhgba29stxwcHBxMWFoa7uztarZaGhgYaGhosXyoeHh6EhoZa9tNuamqipKSE7OxsOjo6LHXNht+zq6uLhoYGWltbLe8RGBhIWFgYMpmMkpISPvzwQ4aGhtixYwexsbGi1Ram1ZSS297enoyMDNavX09LSwuDg4PY29sTHBw87fPK+/v7uXfvHlVVVbzwwgt4e3tP6fy3336bv/zlLwB4e3tz+PBhDh8+jLW1NdXV1bz77rtkZWVZjn/uuec4fPgw7u7u9PT0cOHCBU6dOmXpS4iLi+PQoUOW5M7Ly+NnP/sZeXl5WFtbo1AouH37NuHh4djb21NfX8+7777Lxx9/bHmPvXv3cujQIaytrcnMzKSpqYmAgAC2bt065amqgjAh6QmZzWbLn5nwySefSLt375Z27twpVVRUTPo8k8kkrVq1Sjpz5oxkNBotfz4dp9lslkwm04jXTSbTiGMmet1sNkvf/e53JX9/f2ndunXSpUuXJKPROOF7aLVa6eTJk1JgYKBkY2Mj/eQnP5FaWlqe8l9LEB43qdLGDQ0NaLVanJyccHFxoaWlhdOnT5OZmQnAjh07OHjwIImJidMyBTU/P5+3334brVbL//3f/xEWFjbla8hksjGnf/7tpgpjnT+epqYmqqqq6OvrIyAggNjY2BHnjPUexcXFXL58mf7+fhISEujt7RXbMAkzYtzk7u7u5tatW/zyl79Ep9Ph5OREZGQkANnZ2ZZN6Kqqqjh79iySJLF58+anCignJ4cTJ07w8OFDfvCDH7BixYp5OYGlqKiIBw8eEBgYyLp16/D09JzwC6O3t5dbt26RnZ2Nq6srX/jCFzh16hS7du2alyMBwsI2btY8ePCA999/HxsbG+Lj42lvb6ewsBBra2t27tzJ3r17Abh37x75+fnk5uY+VXLn5uZy4sQJ1Go1hw4dYsuWLU98rZkiSRI6nY6bN2/y4MEDoqKiSEpKmtQXUGlpKXfu3EGj0bBhwwZ27tzJT3/6U/r7+2chcmGpGfcT2dHRwSeffMJPf/pTNm7cSE1NDf/zP/9DZ2cnr732muV2ec2aNTQ0NFBTU/NEQUiShFar5fTp01y8eNGydPTUqVOWY9asWUNERMSUJojMBKPRSEtLC3fv3mVgYIDIyEiioqImPE+r1XL9+nXKy8sJDg5m7969hISEzMu7EmFxGPeTpdPp0Gq1bNmyBVtbWxISEli5ciVDQ0MjnoNdXV2xtrZ+4mdHSZLo6+tDpVLh4+NDR0cH//Vf/zXimFdffRVvb+85T26dTsft27dpaWnBx8eHtWvXWnrQxyJJEtXV1dy6dQu1Ws2mTZvYs2cPVlZWeHt7i9VfwoyYsNmQJAm9Xm95njSZTJjNZvR6/YjjjEbjEwchk8nw9/fn5MmTT3yN2TI4OMjFixdRq9Vs27ZtUkNYJpOJU6dOcf/+fVasWMH27dtZtmwZBoPBsp5bEKbbhMnd2dnJV7/6VUvPc2lpKd3d3bz++usjjisuLl70lViMRiPd3d2WW/LExMQJixcOTzM9f/48arWazZs3WyrYDBdrEJv6CTNh3OT29PQkISFhRF1tZ2dn7O3tLRU7h3l5ebFq1aqZiXKeUKvV5OXl0dnZyerVq1mzZs2E1VJ6eno4fvw4bW1trFu3juTkZMtGBFZWVlOeeScIkzVuci9fvpxvfetbj92Cj0Yuly/6udGdnZ1cvXqVoaEhkpKSCAkJGXfKaG9vL0VFRVy6dAmFQsHOnTuJjY213AUNb0rw6dp0gjBdxk1uHx8fnnnmmdmKZV7T6/U8ePCAW7duIZfLSUlJGXd5pslkoqGhgQsXLvDw4UO2bNnChg0bCAgIsBxjNBo5ceIEX/3qV0VyC9NO7Co3Sd3d3VRUVNDU1ISPjw8xMTHjllJSq9V88sknXLp0CTs7O/bv3094ePiIlt5kMnH16lWUSuVs/ArCEiOSe5Lq6urIzc3FxsaGdevW4e3tPeYtuSRJ3L9/nytXrtDe3k5kZCQ7duyYkeo1gjAWkdyT1NzcTHFxMQ4ODuzfv3/cHU0HBwfJycnh0qVL+Pr68tZbbxEaGiqWdAqzSiT3JCiVSqqqqnj48CHu7u4kJyePW5Thzp075OXlYWNjQ1paGmlpaaNOvlEoFLzwwguTrrkmCFMh5j5OQm1tLdXV1djZ2Vk6xcZqhdVqNVeuXKGoqIiAgAD27NmDq6vrqMcqFAqOHDmy6EcZhLkhknsCBoOBkpISKioq8Pb2Ztu2bTg4OIy5AuzOnTvk5uYyNDREXFzcuHXcZTLZop/4I8wdcVs+gc7OToqLi2lpaSEwMJCUlJRR14mbTCa6u7s5c+YMtbW1REZGsm3btnFbZUmSaGxsRKvVzuSvICxRIrknUFZWRk1NDba2tkRHR7Nq1apRCzkMd6JduXIFnU5HWlrahEtWhzclaGpqmqnwhSVMJPcEbt68SWNjI8HBwcTGxo55XG9vL7/61a9QqVSkpaWxfft2AgMDx7328CSW5ubm6Q5bEERyj0WSJPr7+yktLUWpVLJ8+XJSU1NHPbarq4sbN25w584drKys2LZtG3FxcRNWZoFHCW42m6c7fEEQyT0WvV5PTk4OjY2N+Pj4EBsbO+aQVWtrK+fPn6evr4/U1FRiY2Nxc3Ob5YgFYSSR3GPQ6XSWqaFr1qwhJiZm1J03lUqlpcSUi4sLzz33HBERERPuzQ2PesuTk5PFPtzCjBDJPQqj0YhKpSI3NxetVsvatWtZuXLlY8eZTCbu3btHZmYmKpWK9evXs3nz5knXWLe2tuall14S49zCjBDj3KMYGBigtLSUsrIybG1tiYiIGLGaa5hKpeLWrVvk5OTg5+fH0aNHCQwMnHRdNLlcLlbdCTNGJPcoenp6uHnzJgaDgbi4OEJCQkatczZcpthsNrNx40YOHDgw5YKHkiRNquNNEKZK3JaP4tPJnZaWRmho6GPHDAwMcP36dQoLCwkNDeXw4cOT2t3z04b35xbj3MJMEMn9N/r6+qitraWyshJbW1tiY2NHXap548YNSkpKcHJyYuPGjSQnJ4vkFuYVkdx/o729nZKSEoaGhkhMTGT58uUjVnSZzWbUajXnzp2jurqaiIgIy77cU2U2m6mvr2dgYGA6fwVBAERyjzBcGik/Px9bW1syMjLw9fUdMayl0+nIycnh1q1bWFtbs3HjRpKSkuYwakEY3bzsUOvr66O9vZ2BgQFWrlyJnZ3dlG95n/R979+/T3FxMa6urqSmpo5Yrmk0Gmlvb+cPf/gDra2tZGRkkJGR8cTrsa2srJu95zUAABNASURBVPDz8xt1/FwQnta8a7mHhobIzs7mzTff5JlnnqGurg6DwTAr793Y2Eh5eTlDQ0OsWbOGhISEEUUZ1Go1OTk5fPTRRwCkp6ePO998IsN1y8WmBMJMmHct989+9jOOHTtGS0vLEz3HPo3S0lJKS0txd3dn06ZNjxVkaGlp4Z133sFgMPD888+TmJj4VHXHh5NbbCckzIR51XKfOXOG3NxcYmJi+NKXvjSr7z04OEh1dTX19fV4enqye/dubGxsLK+3traSk5PD3bt3LdVMV6xY8VSPC1ZWVjg6OorNAIUZMa8+VREREbz88svI5XLq6upm9b3LysqorKxEoVAQExPDqlWrLB1per2ekpISMjMz0el0bN++nbVr1z71biEmk4ns7GzWrl076SmrgjBZ86rljouL4zOf+cyc9D7n5+dTV1dHYGAgqampODk5WWaODW9GUFRUhK+vL4cPH8bf339Si0PGI8a5hZk0r5J7LkiSxMDAAIWFhbS3txMaGkpycrLldZPJREFBAbdu3UKv15OcnMyOHTvGLHo4FSaTievXr9PZ2fnU1xKEvzWvbsunU09PD62trcCjBRoODg44Ojoik8nQ6XQMDAwwNDSEyWSitraWoqIitFoty5YtIywsjL6+PgYGBujv7+fatWuUlJQQGhrKkSNHcHNzY2hoiIGBAUtPvkwmw9HR0fIeer2egYEBBgcHLTHZ29vj6OiIjY0NJpOJ/v5+UahBmDGLNrm//vWvW3q7fX19ef311zl69Chubm7cunWL3/72t2RmZgKPWtDBwUFWr17NihUr6Onp4dixYxw7dgytVotOp8PW1paoqCh27dqFQqHg5MmTHDt2jLKyMgDc3d05evQor7/+Oj4+PuTn5/Pb3/6WDz/80BLTwYMHef3119mwYQOdnZ385je/QaPRzP4/jrAkLNrk/t73vkd6ejoANjY2+Pj4WMasExMTCQoK4hvf+AYDAwP8wz/8A7W1tURHRxMTE4Ofnx+vvfYaKSkpfP/736e0tJSUlBSOHDli+cLYu3cviYmJlsqlCoUCX19fy/BdTEwM3/3ud/na175micnT09MyT93T05OXXnoJjUZDcHDwrP27CEvHok3uFStWkJKSMuprrq6uuLq6otPpqK2tZXBwEEdHR+Lj41m9ejW2trb4+vqSnZ2NVqslMDCQtLS0Ec/iPj4++Pj4jPn+Li4u4/amW1tbExISwpe+9CWx44gwI+ZVchcUFFBaWkp9fT2lpaX09/dz/PhxvL292bRpE7GxsdPSkTWsv7+fgoICenp6CA8PJzIyEg8PD4xGI52dnZw9e5auri62bdvGpk2bpn1SjUwmIyoqalqvKQjD5lVveWtrK8XFxZSXlyOXy9m2bZtlIUdzczM6nW7a3stsNtPV1cXVq1fR6XQkJCRYWtC+vj5u377NnTt3cHd3Z+vWrcTExEzbe386hrq6OrEqTJgR86rlPnDgAAcOHJiV9xoaGqKxsZGsrCwkSSI+Pp6goCCMRiONjY2cOHECtVrN/v37SU5OHncv7ic1XLf8xRdfFC24MO3mVcs9mzo7OykpKaGjo4Pw8HASEhLw9fVFqVRy8+ZNMjMzsbW15YUXXpixxBuexCI2JRBmwpJN7ubmZrKzs1EoFKSmplrKC5eVlfHBBx9gZ2fHCy+8QGho6Ig55tNNkiQkSZqx6wtL15JMbr1eT3NzM6WlpVhbW7Nz5068vLxoamoiNzeX8vJygoKC+MIXvoC/v78oYCgsSEsyuZubm7l37x59fX2sXr2a2NhYbG1tuXPnDjdu3MDKysrSiebg4DBjccjlcjZs2DAjz/OCsCSTu7KykuLiYhwcHMjIyMDHx4eHDx9y48YNKioqWLZsGZ/5zGdwcXGZ0QowCoWCI0eOiE0JhBkxr3rLZ4NOp6Oqqorq6mo8PT3JyMhAoVBw/fp18vPzkcvlJCcnk5qa+lixhukml8vZs2fPjL6HsHQtuZa7paWFkpISWltb8fPzIy4ujsHBQc6ePUt5eTnr16/ny1/+8oglnzPJaDSKDjVhRiy55C4uLqapqQlvb2/Wrl2Lra0tZ86coa6ujrCwMNLS0li1atWsxCLWcwszackld15eHnV1dQQHB5OSkkJbWxvvvfcera2tJCQksH79+hkd+vo0g8HAiRMnxDi3MCOWzDO3JEkolUoqKystJZMjIyM5d+4cZWVl+Pr6snHjxlF385zJmBobG8X0U2FGLJmW22w2U1JSQltbGz4+PkRERNDX18ef/vQntFot6enprFu3btYrrgrCTFkSyS1JEnq9nuzsbDo7O1m1ahXu7u5kZWVx9+5dAgIC2Lt3LytWrJjVuKysrAgICMDe3n5W31dYGpZEcptMJlQqFVevXkWlUhEYGIhWq+XDDz/E1taWz372syQlJeHm5jarcQ3XLRfj3MJMWBLP3AMDAxQUFFh6yR0dHWlpaaGqqgpvb29efPFF/P39Zz0ua2trXnrppVnrwBOWliWR3L29vVy8eJH+/n6SkpLo7u6mvLwcV1dXDh48SFBQ0IxPWBmLuCUXZsqivy0fviW/desWQ0NDeHh40NHRQWNjIxEREXz+85/H2dl5ThaHmEwmrl69ilKpnPX3Fha/RZ/c3d3dlJaW0tzcjL+/P729vTQ3N+Po6MjWrVuJj4+fs1ZbrOcWZtKiT+7hBSE6nY6VK1fS0NCASqUiPj6eAwcOYGdnN2dLOoe3ExKbEggzYVEnt8lk4uHDh9y9e9eyuqu9vR0PDw+2bt3KunXr5jhCQZg5izq5Ozs7KS0tpbKyEgcHB+7evUt3dzcbN25k69atcx2eIMyoRZ3cdXV1lJeXI5PJ8PT0RKvVEhUVxaZNmwgPD5/r8FAoFBw6dEiMcwszYlEPhVVWVlJUVITJZKKvrw+j0UhaWhoxMTHY2dnNdXiWSSwBAQFzHYqwCC3a5O7s7KSmpoaWlhZkMhkajYbo6Gi2bdtGaGjoXIcHPNqUYDYXqghLy6K9La+pqaG+vh6tVotMJkMul3Pw4EESEhKmddeSp2E2m6mpqaG/v3+uQxEWoUWb3JWVlbS0tGAymZAkidDQUPbu3WvZiG8+EMUahGH9/f20traiVqsnPFaSJKqrq+nq6rJsIT2aRZvc+fn5VFdXA+Dh4cHhw4dZvnz5vHjWHmY0Gjl58uSCm8Qiaq1Pv5ycHN566y3OnTs34bF6vZ5nnnmG48eP09HRMeZxiza5VSoVGo0Ga2trgoODOXz4sGULX+HpZGdnk52dPddhLCrXrl3Dz8+P1atXT3isXC7n6NGjliq+Y5l3HWoXLlwgMzOThw8fWn72z//8z8TGxk5pkYXJZAJg2bJlZGRkEBAQgFwun/Z4l6Lh2u7D+58Lk1dcXEx+fj5hYWFkZGRgMpno6urizp07HDlyZFIdrDKZjKSkJK5cuTLuI928arkvXrzIRx99hF6vZ8OGDcTFxaFSqfjlL3857jfUMEmSGBoastwy2tnZsXbtWvbv34+1tfW82zlELpeTmpqKl5fXXIcyJUqlckksdjGZTPT09GA2m6ftmiUlJZSUlFimHOt0OgoKCpDJZPj6+o67p/swKysr1qxZg7OzMw8fPqSlpWXU4+ZFyy1JEgMDA/zpT3/CaDRy6NAh9u/fT39/P4GBgfzwhz8kJiaGlStXWvb0Gs3Q0BAlJSUMDQ0BEBoaatnXez4SxRrmN0mSMBqNlJSUYDAYCA4OfqJ1/yaTie7ubv7617/y0Ucf0dXVhUwmw97eHj8/P65fv46/v/+oxUIuX76MUqnEYDAQEhLC+vXrcXBwwM/Pj+DgYDo7O6msrCQ4OPixc+dFcptMJpqbm7l//z579+4lISEBeLTWed++fRw7doza2loaGhrGTe7e3l5Onz6NRqPBxsaGjRs3snPnznnVifZpcrmcnTt3znUYwhgUCgUeHh5cvHiR6upqIiMjSUhIwMfHBz8/v0lfx2g00tbWZrkD9fT0pLe3F6VSSXp6OkVFRaxYsWLE1lV6vZ6GhgZ+/vOf097ejtlsJiYmBicnJ5KTkwHw9vbm/v37Y7bc8+K23Gg0UlVVhVarJSgoiMDAQODRh9/f35/AwECMRuOE48H9/f1cunSJ/v5+goODSU1NJS4ubjZ+hSem1+un9bZPmF5yuZxXXnmFVatWcezYMQ4fPszvf/97ent7GRwcnNT/O1tbW6Kiojh58iSRkZF8/etf5/Lly5w+fZr9+/fT3t6Om5vbiOTu6OjgG9/4Bt7e3vz617/mypUr7Nu3j9OnT1uO8fLyQpKkMYfP5kVyTxcvLy++/e1v4+npybPPPktKSspchzQuo9HI+++/P6LzUJifPvvZz/LGG2/g6+vLj3/8Y2JjY/m3f/s3GhoaJnW+2Wyms7MTg8GAu7v7YxOpvLy8Ru0wbm9vR6vV4uLiwp49e/h//+//WV7z9PQcdwRoXtyWTxcnJyf27NnD//7v/3L+/Hnu3Lkzo7t0Pi1Jkmhra8Pd3X1BlVuqqqoCoKKiYo4jmV1dXV20tLTQ19eHRqPhj3/8I4WFhRw9epR9+/aN2xlmMBgoLy9n2bJl+Pj4PLbBpFwuH9Hh6+3tzY9+9CNeffVV3nzzTVJSUti/fz+7d++2HCOTycbtJF5Uya1QKPDx8eE73/kOXV1dluEwQZgO9fX13Lx5E7VajaOjI6tWrSI1NZXw8PAJi1waDAYKCwsJCwvDx8fnsdf7+vrQ6/WWv9vZ2REXF8c3v/lNmpqaKCoq4vz586xdu5agoCAANBqNpfN4NPMiua2srHB0dEQul9Pb20tfXx8uLi6YzWb6+/vp6+sjKCgIZ2fnSV3rhRdemIWohaWkoaGBc+fOUVNTQ3p6OvHx8aSmppKSkoKvry8KxfipZDAYqKioICkpCUdHR8vPZTIZrq6uqFQqtFot8Gj/+OLiYp599lkOHTpEc3MzTU1N1NfXj+h3Gp5+OtZd37xIbrlcTnBwMC4uLtTW1lJdXU1SUhIGg4GSkhK6uroICAgQSyOFWSdJEp2dndy5c4fKykqWLVvGrl272LZtGw4ODlOaO2E2m3nw4AF5eXnIZDJCQ0OxtrYmIiICpVKJRqMBHpUGe//99y1fAs3Nzeh0OqKiokbMiWhtbUUmk416JwDzJLkVCgWrV68mNjaW8vJyrl69SkBAAL29vfzxj3/E09OTmJiYKQ0/CMJ0MJlM5Ofn093dzaFDh0hKSnqi/hFbW1vS09M5e/Ys165d47nnnuPNN9/E19eXdevWceLECZRKJWazmVWrVvHss8/y8ssvWyZkHTp0iL/7u7+zJLfBYKCqqgpPT88xlzBbSfNoBYBWq+XNN9/k3XffxcbGBrlcjpeXF2+//TaJiYlzVqVUWJpMJhP9/f2WW19bW9vHOr4mS5IkdDoder0eSZKwtrbGzs4OvV5PQUEBR48e5Wtf+xovvfQSbm5uGI1Gy206gI2NDba2tshkMsu+d1//+tfZuHEj//RP/zTqasd50XIPc3Bw4F//9V85evQo8Oj52cbGhsjISJHYwqyTyWSWoaaJeqYnYmVlhZ2d3WMTqmxsbIiPj2fZsmWoVCqUSiXu7u5YW1uPWXdAkiTu37+Pv78/UVFRY07smlfJDRAWFkZYWNhchyEIWFlZzfhio+EvkGeeeYbi4mKKioomXDxiNpvJysoiKiqK2NjYMRu+eZfcS0VfXx+5ublUVlayfft2IiMjsbW1neuwHjMwMEB1dTV//etfLT/z8vIiISGB+Pj4OYxscdmzZw8eHh6W2ZnjkclkpKSksHbt2nELfYrkngNtbW1kZWXxzjvvcPPmTX7xi18QEhIyL5O7sbGRc+fOcevWLezt7dHpdPT19VFXV4ePj8+kPozCxCIjI4mMjJzUsXK5nC9/+csTHieSe5b19/dz+fJlfvCDH6DX68ctkzMfNDU10d7ezu9//3v8/PxQqVR873vfIzc3l5UrV/Lyyy/PdYjCGBbV3PKFICsriw8++IDQ0FB+97vfzfu13Hv27OEXv/gFgYGByOVyfHx8SE9PJzAwcMlNP11oRMs9y7Zs2UJcXJylIN58Z2Vl9VgvsVqtxmg0zvsvpqVOJPcsc3FxwcXFZcEk99/q6emhsrKSgYEBIiIi5jocYRzitlyYkqysLJqbm1m+fPm8Xyu/1ImWW5gUo9FIWVkZ58+fx8fHh3379s2bnVuE0YmWW5iQTqejrq6On//856jVap599lkyMjLmOixhAqLlFsZlNptpbGzkRz/6EefPn+fs2bNs3rx5rsMSJkEktzCuwsJCfvWrX3H16lUuXbrE2rVr5zokYZLm1aqwpeDOnTucO3eOu3fvolarKS0tJTIyEl9fX1JTU3n55Zfnzdx6jUbDu+++y/e//326u7tJS0sbMYsuIiKCN954g4iICLHhwzwkWu5Z5u3tTWJiIt7e3gB84QtfsLwWHh4+qWozs8Xa2pqEhAS+/e1vj/q6r68vrq6u826zB+ER0XILwiIlessFYZESyS0Ii5RIbkFYpERyC8IiJZJbEBYpkdyCsEiJ5BaERUoktyAsUiK5BWGR+v8dZMCrz0IFEwAAAABJRU5ErkJggg==)
Two tuning forks
and
are vibrated together. The number of beats produced are represented by the straight line
in the following graph. After loading
with wax again these are vibrated together and the beats produced are represented by the line
If the frequency of
is
the frequency of
will be
![](data:image/png;base64,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)
If a hyperbola passing through the origin has
and
as its asymptotes, then the equation of its tranvsverse and conjugate axes are
Here we used the concept of polar coordinate system and also the trigonometric ratios to find the solution.
If a hyperbola passing through the origin has
and
as its asymptotes, then the equation of its tranvsverse and conjugate axes are
Here we used the concept of polar coordinate system and also the trigonometric ratios to find the solution.
Five distinct letters are to be transmitted through a communication channel. A total number of 15 blanks is to be inserted between the two letters with at least three between every two. The number of ways in which this can be done is -
Five distinct letters are to be transmitted through a communication channel. A total number of 15 blanks is to be inserted between the two letters with at least three between every two. The number of ways in which this can be done is -
The number of ordered pairs (m, n), m, n
{1, 2, … 100} such that 7m + 7n is divisible by 5 is -
The number of ordered pairs (m, n), m, n
{1, 2, … 100} such that 7m + 7n is divisible by 5 is -
Consider the following statements:
1. The number of ways of arranging m different things taken all at a time in which p
m particular things are never together is m! – (m – p + 1)! p!.
2. A pack of 52 cards can be divided equally among four players in order in
ways.
Which of these is/are correct?
Consider the following statements:
1. The number of ways of arranging m different things taken all at a time in which p
m particular things are never together is m! – (m – p + 1)! p!.
2. A pack of 52 cards can be divided equally among four players in order in
ways.
Which of these is/are correct?
The total number of function ‘ƒ’ from the set {1, 2, 3} into the set {1, 2, 3, 4, 5} such that ƒ(i)
ƒ(j),
i < j, is equal to-
The total number of function ‘ƒ’ from the set {1, 2, 3} into the set {1, 2, 3, 4, 5} such that ƒ(i)
ƒ(j),
i < j, is equal to-
The number of points in the Cartesian plane with integral co-ordinates satisfying the inequalities |x|
k, |y|
k, |x – y|
k ; is-
The number of points in the Cartesian plane with integral co-ordinates satisfying the inequalities |x|
k, |y|
k, |x – y|
k ; is-
The numbers of integers between 1 and 106 have the sum of their digit equal to K(where 0 < K < 18) is -
The numbers of integers between 1 and 106 have the sum of their digit equal to K(where 0 < K < 18) is -
The straight lines I1, I2, I3 are parallel and lie in the same plane. A total number of m points are taken on I1 ; n points on I2 , k points on I3. The maximum number of triangles formed with vertices at these points are -
The straight lines I1, I2, I3 are parallel and lie in the same plane. A total number of m points are taken on I1 ; n points on I2 , k points on I3. The maximum number of triangles formed with vertices at these points are -
If the line
is a normal to the hyperbola
then ![fraction numerator a to the power of 2 end exponent over denominator l to the power of 2 end exponent end fraction minus fraction numerator b to the power of 2 end exponent over denominator m to the power of 2 end exponent end fraction equals](data:image/png;base64,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)
So here we understood the concept of hyperbola and the normal lines.In analytic geometry, a hyperbola is a conic section created when a plane meets a double right circular cone at an angle that overlaps both cone halves. So the value of .
If the line
is a normal to the hyperbola
then ![fraction numerator a to the power of 2 end exponent over denominator l to the power of 2 end exponent end fraction minus fraction numerator b to the power of 2 end exponent over denominator m to the power of 2 end exponent end fraction equals](data:image/png;base64,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)
So here we understood the concept of hyperbola and the normal lines.In analytic geometry, a hyperbola is a conic section created when a plane meets a double right circular cone at an angle that overlaps both cone halves. So the value of .
If the tangents drawn from a point on the hyperbola
to the ellipse
make angles α and β with the transverse axis of the hyperbola, then
So here we understood the concept of hyperbola and the normal lines.In analytic geometry, a hyperbola is a conic section created when a plane meets a double right circular cone at an angle that overlaps both cone halves. So the correct relation is
If the tangents drawn from a point on the hyperbola
to the ellipse
make angles α and β with the transverse axis of the hyperbola, then
So here we understood the concept of hyperbola and the normal lines.In analytic geometry, a hyperbola is a conic section created when a plane meets a double right circular cone at an angle that overlaps both cone halves. So the correct relation is
The eccentricity of the hyperbola whose latus rectum subtends a right angle at centre is
The eccentricity of the hyperbola whose latus rectum subtends a right angle at centre is
If r, s, t are prime numbers and p, q are the positive integers such that the LCM of p, q is r2t4s2, then the number of ordered pair (p, q) is –
Finding the smallest common multiple between any two or more numbers is done using the least common multiple (LCM) approach. A number that is a multiple of two or more other numbers is said to be a common multiple. Here we understood the concept of LCM and the pairs, so the total pairs can be 225.
If r, s, t are prime numbers and p, q are the positive integers such that the LCM of p, q is r2t4s2, then the number of ordered pair (p, q) is –
Finding the smallest common multiple between any two or more numbers is done using the least common multiple (LCM) approach. A number that is a multiple of two or more other numbers is said to be a common multiple. Here we understood the concept of LCM and the pairs, so the total pairs can be 225.
A rectangle has sides of (2m – 1) & (2n – 1) units as shown in the figure composed of squares having edge length one unit then no. of rectangles which have odd unit length
![](data:image/png;base64,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)
Here we used the concept of number system and the rectangle, we can also solve it by permutation and combination. herefore, we get the number of rectangles possible with odd side length = m2n2.
A rectangle has sides of (2m – 1) & (2n – 1) units as shown in the figure composed of squares having edge length one unit then no. of rectangles which have odd unit length
![](data:image/png;base64,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)
Here we used the concept of number system and the rectangle, we can also solve it by permutation and combination. herefore, we get the number of rectangles possible with odd side length = m2n2.