Question
A is a set containing n elements. A subset P1 is chosen, and A is reconstructed by replacing the elements of P1. The same process is repeated for subsets P1, P2, … , Pm, with m > 1. The Number of ways of choosing P1, P2, …, Pm so that P1
P2
…
Pm= A is -
- (2m – 1)mn
- m+nCm
- (2n – 1)m
- None of these
The correct answer is: None of these
Let A = {a1, a2,…..an}.
For each ai (1
i
n), either ai
Pj or ai
Pj (1
j
m) . Thus, there are 2m choices in which ai (1
j
n) may belong to the Pj
s.
Also there is exactly one choice, viz., ai
Pj for j = 1, 2, …, m, for which ai
P1
P2
...
Pm.
Therefore, ai
P1
P2
….
Pm in (2m – 1) ways . Since there are n elements in the set A, the number of ways of constructing subsets
P1, P2, ….. , Pm is (2m – 1)n
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will be
![](data:image/png;base64,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)
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==)
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.