Question
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,iVBORw0KGgoAAAANSUhEUgAAAFwAAAAqCAYAAAA+lDyEAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAbSkFbcgAAAkNJREFUeNrtmjFLA0EQhRcRCRaChQQJIoiI+AdELGysLCSdWKaxsLQJEoKIvYWdWIiFCEFSiV2QFHYiYimIlUgQLERCEOGcyU3gOPfgLiHnG90PHniDxdt1bnfPfcb8XzzRJ+maNO1Mp8MAaZN060ynS8uZTo8lUt2Z7p6mvHVxGJflcErTZCOZ5n3kPubvzpAuxL8a0EznSWcxm+SSNIK0Ee6SGnIKuSNNopsmdkgl8f4i+8qVpSHY9yxS53LXHpBGZfIr0j3QpsXnI6kY8L5v6XrPol9jXYwHOSGtIJsWHkhzoRq/ge/I6zK/gguhWt2ypCB+DzQt9Qz6hIePVfzzh4INfF6aJQw3Tw3Z+GvoeVHJVyQvhceW+hapjGz8WV7DDmz2XMGE8yZfCNWG5FwOfdbeIx3KUsLr9qmcSNCpyqa5LM85qRU0fNCU5KRSlm5vWI6FaLDHVZn0L/l2yBuljBmHw+Fw9BEvhhya/lrdqB/d4aUsh1tSHA6Ho+e1S3MASOVY/kIASOVYWn/o7YUfi8rUkoax2NJMKlNLEUCNxZZmUplaigBuLOE0E2IAqJfOhhsLp5mKgWe0AFDHX9b495p888Opq06OhgNBHAZ6M35cYht4LG3CqSu0/xlUZNJvjH9zP2j8u9gn6WDeCNekPiEnkSzoWNrwHWEOeFlgfzXp5CCcPjiKqMPuPVFpJjR/wwnrsESlmdD9Ja3DEJVmQveXtA4D7/obCv0lrcNQNT9jyhr8Ja3DwGfXjEJ/Seup8A0wjuluqiH/JQAAAPV0RVh0TWF0aE1MADxtYXRoIHhtbG5zPSJodHRwOi8vd3d3LnczLm9yZy8xOTk4L01hdGgvTWF0aE1MIj48bWZyYWM+PG1zdXA+PG1pPmE8L21pPjxtbj4yPC9tbj48L21zdXA+PG1zdXA+PG1pPmw8L21pPjxtbj4yPC9tbj48L21zdXA+PC9tZnJhYz48bW8+LTwvbW8+PG1mcmFjPjxtc3VwPjxtaT5iPC9taT48bW4+MjwvbW4+PC9tc3VwPjxtc3VwPjxtaT5tPC9taT48bW4+MjwvbW4+PC9tc3VwPjwvbWZyYWM+PG1vPj08L21vPjwvbWF0aD4H3Vu/AAAAAElFTkSuQmCC)
Hint:
A hyperbola is the location of all the points on a plane whose distances from two fixed points in the plane differ by an amount that is always constant. We have given that the line
is a normal to the hyperbola
, we have to find
.
The correct answer is: ![open parentheses a to the power of 2 end exponent plus b to the power of 2 end exponent close parentheses to the power of 2 end exponent](data:image/png;base64,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)
A hyperbola is a significant conic section in mathematics that is created by the intersection of a double cone with a plane surface, though not always at the centre. A hyperbola is symmetric along its conjugate axis and resembles the ellipse in many ways. A hyperbola is subject to concepts like foci, directrix, latus rectus, and eccentricity.
We have given: the line
is a normal to the hyperbola ![fraction numerator x to the power of 2 end exponent over denominator a to the power of 2 end exponent end fraction minus fraction numerator y to the power of 2 end exponent over denominator b to the power of 2 end exponent end fraction equals 1](data:image/png;base64,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)
![N o w space e q u a t i o n space o f space h y p e r b o l a space i s space colon
x squared over a squared minus y squared over b squared equals 1
T h e space e q u a t i o n space o f space n o r m a l space w i l l space b e colon
fraction numerator a x over denominator s e c theta end fraction plus fraction numerator b y over denominator tan theta end fraction equals a squared plus b squared space...................... left parenthesis 1 right parenthesis
T h e space e q u a t i o n space w e space h a v e space g i v e n space i s space l x plus m y minus 1 equals 0
S o space w e space g e t colon
fraction numerator 1 over denominator begin display style bevelled fraction numerator a over denominator s e c theta end fraction end style end fraction equals fraction numerator m over denominator begin display style bevelled fraction numerator b over denominator tan theta end fraction end style end fraction equals fraction numerator negative 1 over denominator negative left parenthesis a squared plus b squared right parenthesis end fraction
fraction numerator l s e c theta over denominator a end fraction equals fraction numerator m tan theta over denominator b end fraction equals fraction numerator 1 over denominator left parenthesis a squared plus b squared right parenthesis end fraction
s e c theta equals fraction numerator negative a left parenthesis negative 1 right parenthesis over denominator l left parenthesis a squared plus b squared right parenthesis end fraction
tan theta equals fraction numerator negative b left parenthesis negative 1 right parenthesis over denominator m left parenthesis a squared plus b squared right parenthesis end fraction
N o w space w e space k n o w space t h a t space s e c squared theta minus tan squared theta equals 1 comma space s o space w e space g e t colon
fraction numerator a squared over denominator l squared left parenthesis left parenthesis a squared plus b squared right parenthesis squared end fraction minus fraction numerator b squared over denominator m squared left parenthesis left parenthesis a squared plus b squared right parenthesis squared end fraction equals 1
S i m p l i f y i n g space t h i s comma space w e space g e t colon
a squared over l squared minus b squared over m squared equals left parenthesis a squared plus b squared right parenthesis squared](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 .
Related Questions to study
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
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5E8L1Kkm2AQLJJaPUECdCOprIC5GfcweUTW+Bjb4llR+8is0osi5Bs2gvJJqHlstUh88Rq+Iz/Aj+OMoXxTH/EZlaiqa37dU2S7UDAPjHzYdzdlsJ1ySLxkXqk3r4F47EGePL4kZh5P5WVlbAwN8GJ8FAxM/jIfPpEaLOU5Jti5uOy3N2VZJNHhbriTDxJvohrV2OQsM8P8+f648itF1L3kqNLtqDA/WLmw/RHtjupt4UT5+mTx2Lm/VRVKQTZIk6EiZnBx7PMp0Kb3UpJFjMfF08PNyx0nis+0ixaLltPOjJD8fMEEyzddxH3a8UkR2NjI+xsJkNfdwRcFjmrFfyxejrDZWt9hb3tFAz55t9wdLCTrfeOubNn4IfvvsJEi/Gy9cEQ07m24tvMbqq1bF2TsXjhfO71+Borli8TzxzNon2ytZai/MFlhEfcQ0Z5I7p/jK2sSUKk47dY5BuOuDdikqNLtrH6ukKfXJ0w1NOBweiRsrW+4u2J8yVmOf0kW+8d/Dvoj0O+hrWVhWx9MMSsGY5CmzlOs5WtazKWubpg6A/fclc3ku0tylxkHfWA2fd2WH2xANJciArK54FwH2OMxVsS8aDbla2rG3kwKFDMfBiPZa5wW7pYfKQe9+7eEbpEfNdIHWpqqmE5wVT4nHCwkp31TGgzfryrDfBXNepGdtHZgZrcBIS5msNnVzTOZ+SjoICLrJu4c3QVpi4KQPDtV2jvNkfSJVvg/r1i5sMsc12CpS4LxUfqcftWinDiPH6UIWbeT2VlhTBmCw87LmYGH/z4lm8zfiZXG6AJkl50tivR8OoaolY5wMloGAwMRkFvnBWmLDqAqMdlqGvpuUaSpv61F5r6l9DiCZJOVGXeQFLsCZw+fRKRZy7g4r03aBSr3SHZtBeSTUKLZVMfkk17IdkktFi2OhSnJOB8yBEcDwvBsaOnEXMhG2VitTskm/ZCsklopWztqgqUZoZgi40lLH/UgZGpPsaOMob5WA/suvoMLxpae3wkQLJpLySbhPbJ1lmP8kdB8DPRw2SXw4i8/xpvSrh4dh0X1tvDwnAZtl7IQaV4OA/Jpr2QbBLaJ5syC5lHXGD4hSO84rpL1YyazEB4jbDFyn03kSkz9U+yaR8km4T2ycZ1ISsfJyP2bAayq5u7baVpRvOLk/AznITl2y8jnWQbEJBsElo8QdIL5VOk7nKAkckibInJRinJNiAg2SQGhmz1Rbgf7osFUywxbVsC7r1WiYW3kGzaC8kmoeWyqdCkKMKdY75YajsJNtuuIKtC1WOXNk+XbEEHAsTMh+HXRroyro28eydVWHrE79FSh+rqKnGLzeBdG5n1LFNos1spKWLm40JbbGTpQKfiKkK9p2Ki5RysCM1AblXPK1oXXbLt3bMTbW1taoXrksXCtgu5Wl/BvzuPM9RD2v17svXe8erVS5iPN8Kxo8Gy9cEQ6Q/ShDa7euWybF3TwS8+pytbD5qAzDBsXPQTHBavxd7LD/G868Y/MjQ1NcF+qjWGDvkGDvZT1YqRw77HiKFDZGt9hYWZCb758jNYWZjJ1nvHVGsrfPf1FzAaqy9bHwxhZWkmtBnfnZarazTsbPDlF5/Cy9NDPHM0ixbKVo/a5wmI9JiDOe77cSK3Rcz3Db+fzdZmEizNTbFpg59aYWI0RujeyNX6Cn7z4Y+c0Py7o1y9d/j6eGE4J/T0n+xl64Mh+D1kfJvxVxO5uiZj4/p10NMZIaz8/xhon2xNT5AWYAfd//s9Jrpux4GoCJwKOy5sUwk7HoKw0ONITC9GSbN4PAffjZw00RzBhw+KmQ/DbyDkB8ss8F0ik3GGyH2eI2beD/8mwO/Sjjp9SswMPvLzcoU2u3/vrpj5uHivWE5jtndUPMLdoDmYNtkERsZGQtdtwnhjIcYb6cPM1Bgeh2/hQbV4PEfXmC1gn7+Y+TD92c/G30eDn1njZ9jUoby8TOg+hYUeEzODD/7mSHyb3Uy6IWY+Lh7LltKY7R3tKrQ2VqOqqhIV5aUoKy0RptCFKHn7VVHfjO5b2mjqX3uhqX8JLZ0gYYNk015INgmSjQGSjR2STYJkY4BkY4dkkyDZGCDZ2CHZJAaGbKoGNFZVoK65s8ddtbog2bQXkk1iAMimQsHFbQhYsRCBqUpUyqzYItm0F5JNQstl60TJ1X1YZ/S/oas/Hr6JTagg2QYUJJuEdsqmqkHF/aM4sGUJbMyNofvJf2C8jS02Xm+mK9sAg2ST0ErZOqqK8GzfbMyfaw2r5buxym061jhbY8v1JpS1igd1g5eNX64VcuSQmPkwnsvdhYZngZeMX3qUl/tczLyfJmWTuFwrUswMPgry894u17p/T8x8XHy8aLlWD9oVCrxOTER6RSX4W/q/iFmFLTOssOlqo6xs9fX1mDLZEl6e7sL2F3XCydFB+Gs0crW+4sjhg8JCVv7vrcnVe0di4gWMNRwtLIKVqw+GOHkiXGizQ0EHZOuaDF54/g98sC7T+6PQzm5kRwc6WlTvbldXEOv7Xtn4LTbTbKfg80/+LnRZ1An+zxjxIVfrK3RHDcOXn38inDxy9d5hqK+Dr774FCOGfi9bHwyhpztCaDPdkcNk65oMfl/dP/72n9zVzVM8czSLlk+Q8KiQE70Sm9W4svGLTPnFwuoEf1VzsLeRrfUVBwL3Y/So4cJmULl67zh3Lg5jDHTht261bH0wROixEKHNAvbvla1rOvg3ZdY/gvlH8ZeQrT9jthX8mI2Tk4WH6elMYzalUimM2aKjBvOYLV9oM74bpw3QmO29qCcbzUZqJzQbKUGyMUCysUOySQwI2bIjPbF+mhn8LjeilGQbUJBsEgPjynbGF1tmTsbma3RlG2iQbBIDQLZONJbmIO9RGvIq23vs0O6CZNNeSDaJASDbhyHZtBeSTYJkY4BkY4dkk/hLycYvCVKX5e5uwj0NWeBvx2Y81hBZWc/EzPupq6uF5QRTnDp5QswMPp7nZAttdif1tpj5uPBL+uhztt8BLxt/Um/ZtAEKhUKtmD93NubMcpKt9RXnz50Vlmpdv3ZVtt47srIyMW6MnrB6Qq4+GCLpxnWhzeJiY2Trmo75c2dhwfw54pmjWf4SsvFrI/nlWv/+7J+YOGG8WvHd1/8Wbg0uV+srxujr4rNP/i4IJFfvHWYm4/DFv/4BnRFDZetaHRZmsLI07xbcYwuZ4z4QfBeSbzN+nahcXe3gf55+/P/vgvv5+ef45P/9H+EGvR+Dv4Rs7e3twk1Ag48cQvBh9YLv2vEhV5MN7rlDjx0VtsscPxoif0yvOBp8RLgbMn83Z7m69sYRHA8LR+SZKJyJ5uM0Ys5E4GRYCI7KHt938OtI+Tbg206urlYEhyAs4hROhh9DaLBM/QMREnIMEdzrFsX9LvwV9sWLfPHM0Sx/CdmIP4GaTNwODuLkCsKRw4dx+PBNPC4Xax8BZdZ1ZBWUC1uu2KlHYfwpRB4IxJGQg/DffQJhcdk9/i67JhiUsjUUPUJBfjHKuv29gL7pREPxY6TFn8WFC+eREH8RiZefobhB5gO/LhpLUZR6AZcSziE+PgEX0ovwqlGsaT1KVOclIGL1HNgPM8FUrns+xXoSLIZNxjyvIzibreD/xpAGqcerG3uxceJnmLnuJC6Wimm16EBLzTOkR6/B0rHmsB5nhsk2FrDUm4ApE72w+0YOipTioRpgkMnWgabXSTixWB9uPwfgTLGY7pNWNFXewOlVs2H79VCMHasDw9GGMPhxLtaG3UZ2XQvaxCPf0aJA5vmt8JnwI0wNRsLAYBS+s1+FX849R53M6hftohNoTkao83iYGyzAhivlaGhWorm5GYqkzfAyt4LNgqNIbqhDZakCjbWVKCspRu7z58h/WYbyehXXZDVQvCnE85y3ubLGdvG5GVHVoaasEOlXDmGd2WcY+en/xKSN53GD5era/gqPT3li5neGmL4zGQ9L+d9FCWX+RUT7TIGJvg+C0vLwsqIadYpK1FaVCH8IJK+gCEUKJdpVSrRWvxJ2efC54iql7KIKdRkksnWio12F2lcpOLl0DEz++d8xzu0QYt8rWxvampIQtoB7F7TywIbIx8jJyUbOszQk+8+Do/EsLN2bgkLx6Ld0oDRxA1YunABrvxg8fJbFfU8qwn2nYp6zF3Yl16Ofp56GaAGeBWHFBHtMXBKBtB4/bDrOrFgED6sF2HUqCJ72rjiwfTXcPWZCf9i3GGU1G9N3X0bhtQBsd5uCET9+i5GTnTHncAaqlDL3H/wAqicnEOI5DiPHGGOE0RQ4mXyFpXvicIXlylaTjIt+c2Cg54tDOTX8byeiQMHVnVgxwg7bDvvDd9lq/OLqiR27PDF+nC73BmkIXY9QXEw8i5TAhRg7ejj0xhpjjE8krub2v4syOGTjrmaX/GdzXaLxGDZ8DMxG/B2zNwQj7n2ytVdBeWcTpn1tgoleZ5DR/cSrj8H2iTaYvyAYN991Q7i3vIY7iHCxxERLF+x8Jr0FNl/ygqv9FFhvScWb3/HO+OfD/ZJVeXh05zEe5FVx1/XuZCDGZxHcrJywab8PJnz1A0bY+WBDeCLORGzC5rkG+JuuFSYu34XgU+dx+tSvWONsh/GTNiAmu57rDLJRe+kEYvzXYFv8FZy9GoNgZx147YjExRLxAHVoKUNJZgaS7xWjor17w1ei6NoeLNedjI171mDeZBP8MMQKM3fH4OyZ44hZb4NRBvr4cqonNh2KQWz0UUTscIaZ4RysOZaGF+KzsDIoZFPejkL0uhlYsM0fW0MjsG/+aPhsCEBUkXiAHB2NUOXeQOTxK7j8tLzXiXcXx5zssXDWHsTXiSm+C6bMxt3YGMSdfYg3YpanOnYJFtvbwnHfE1RqtWx90Y6yi75wnjQVM38OxJWolXAc9Q103GJxQ8HX6/A6zgVWX32OkT43kC68+dch+4Q75g+zxNqEUrz6TX/7/SgLX3JdVVH41kc4t1QPy7dEsMnWB20vz+K420QY2K5D9PkD+HXmSAwZ4wK/FPHFqT6FjRM+wb+MvLEt/W0K1VHYOnY05qw8jeus7xwig0K2ppxcvMx8jgrhUTGueY+Dp+8enH6fbH3SiAruxJtlZoPZ6xPwrI9Jls66QjyJP4aTIVvgameNhasCkfhaLA4omlByOwhrZlhgklsAIh5moSRxNebrjYEb10V8JvTNavEieQfW6ppjfWQh8oTbDVbiSfRyuI8whPfJEhSqNRklT3NFKqJcRmP51pO/W7bWF8k4tc4RU+xdsCYmA+UFUQiYbgibmdsQ2/Xc7UkItjPFsiUHcbFKSKClOhb+5j9izrJQxPfzZxhkEyScBC3ZSPAcC89V/v2QrQFl6eH4dZYpzBZtx5H7NWL+tzQUJyHSfQKmmY6G7kQ3+AZcQ0ZRLTQ4+fW76awvwtNrR/DrHEtYLgtA2GP+OlOKFwnrMVvXAWsjM8UxayXyru3Far058L9UjLfvKeV4HOkBt5HjsDKqBEWqZtQUZeHJrVu4e/cO7v0mUnEn9R7up+XjTX3Piac/RrZalGTfxMk1TnBwWAiPqBfc/8F1mxUx2Gpjg5mLD+G6cEXmcs3XEWTjhJU+J3FbeONQobnyDHabDcNcj3BcKONz7Aw62dqVWYjvh2ydLfV4eTcMu+eYwthlHyLSFWjuMYHQi/YWNFVXoLK8HOWPI7Bn4WLYLQpBSksn/3JqOe1QNT1F6tFlmGFmjp82xiPpRTNahXmOlyg47yfK9hQFwvG8bP74WZCtCK+EXJdsJvg5pgSvO1/ijr8b5v7wAwwN9TB2TO+7X42E/khjjDfbhLBHJT3GeL9btjYFFA8PYf1PxjCfsR77blaCnzgVxCo5LSPbNZLtj6B/suUh7dhyzLa0gtO6SMRnVqORaexVgdQ987HEbgb2pLehTtttUyQhZvM8TLJdjE3R9/GwTMWPSEVeolCQbRrWnOot22zsSewtmxF8uStbsUqJ6sKnyEhKwu3bt4RI7RHJuJXMXd3u5OJVXfMfd2VTvcKb+I2YP3UaFmwJxfmcMlS/e+24/+UNL9sUzFh0ENca+FyXbNPh4x2BW8KgkWTrF8yyVacgevsSODrMgfeBOFwr7MsULl98EeHbDuLXsJ4TJPwHsxkHZ8Hb1ByrL6mg0FrZuLNQ+RhXtnvA1WkFNl8qlFmxocCbi36Yo2uP1WrJxnUjT5eAuzD2m/7LVor8K4H4xXEWluxKxF25WXvFGWybag2nhSTbH476srVxY5YspPovx5IZy7HmXA7eTTzKwr1IrxOww3YczO38cLb7C9txGyd8f8IEy9WIyutAPz520gwdLWi8vx6LRn2Kf307GW679sJ/x1Zs//VtbN20AYfD9+DoHh9M+tIKPmEZyBW+sRw5ib/A/Zup2HquAG8/USlFeqgz5n7+I1zDXiP/d8imLLuJ8JlfY8GaYzjLMslUmciNm3Xwr//1A6w9tuBX/+3YKf4uv2zeiN17tiEmdicWGljCfpY/EoW+K/c6KhOxx2QSli45ihuibMryCGzR+RR2zsGIowkS9WjnxiKxLsPh4rENJ3p+It2T9ipU3fwZjkOGwHDqKuxPTMSlBH651jnEnz+Lc7HRuJ6WjaIe08AtyD/3Cw6sXIj1gXGIPReLc+fOIi54I1au2gaf0MfcdUGL4WSrSQ2Ev+902Ey1gYOtNey4d/2usLY0h6uvN3aHRGDzsu04mfICb8+7OrzJiMUxjx2IfVDOXed4alB4MwiBbt4ITq4C1xPtN621WUje54GDp24gTZgdVJPim7gSsAAznOwxjfv5+ej6XaZYWWD6jJ/w6/FQ/OK7C3sDE5AhzF5xV/fWDCRs+AXBR24gU+jPtqG17hZiVi3FzqBruF/N59gZfLI15+HaJgds2nYU8e97l1SWoeiMO+ZP1sWwkaNhajwGpuMMYMKFMTegN9T5AdNXH8bZ3sJ2tuJVSjB2OujAzHAUDAz0oGPli19js1HLNM77OHR2tKGtTYXW1ha0NDcLS7XehVKJltZWqNrauWPa0dHR+W4s19nZgXY+19kt19HO5drQ3u24/sGvAOKfp4P7f8SUOnA/Uwf3u6hUrWhp+e3vwn9tFX+X9nbuucVvE/6/3+T45+J+Bi7X35dx0MnGy9CkeANFVS0a3/dBK3fStdaWouT1C+Tn5yLveQ6e53RFNnKyn6HwdSVqpDVAEqp6VL3IRm4Ov1wrG9kFZVA0DQDTiD+VwScbQXwkSDaC0BAkG0FoCJKNIDQEyUYQGoJkIwgNQbIRhIYg2QhCQ5BsBKEhSDaC0AjA/wdHnqVUtvtHqAAAAABJRU5ErkJggg==)
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.
nCr + 2nCr+1 + nCr+2 is equal to (2
r
n)
nCr + 2nCr+1 + nCr+2 is equal to (2
r
n)
The coefficient of in
is
The coefficient of in
is
How many different words can be formed by jumbling the letters in the word MISSISSIPPI in which not two S are adjacent ?
The different ways in which items from a set may be chosen, usually without replacement, to construct subsets, are called permutations and combinations. When the order of the selection is a consideration, this selection of subsets is referred to as a permutation; when it is not, it is referred to as a combination. So the final answer is .
How many different words can be formed by jumbling the letters in the word MISSISSIPPI in which not two S are adjacent ?
The different ways in which items from a set may be chosen, usually without replacement, to construct subsets, are called permutations and combinations. When the order of the selection is a consideration, this selection of subsets is referred to as a permutation; when it is not, it is referred to as a combination. So the final answer is .
The value of 50C4 +
is -
The different ways in which items from a set may be chosen, usually without replacement, to construct subsets, are called permutations and combinations. When the order of the selection is a consideration, this selection of subsets is referred to as a permutation; when it is not, it is referred to as a combination. So the final answer is .
The value of 50C4 +
is -
The different ways in which items from a set may be chosen, usually without replacement, to construct subsets, are called permutations and combinations. When the order of the selection is a consideration, this selection of subsets is referred to as a permutation; when it is not, it is referred to as a combination. So the final answer is .
The number of ways of distributing 8 identical balls in 3 distinct boxes so that none of the boxes is empty is-
The different ways in which items from a set may be chosen, usually without replacement, to construct subsets, are called permutations and combinations. When the order of the selection is a consideration, this selection of subsets is referred to as a permutation; when it is not, it is referred to as a combination. So the final answer is 21.
The number of ways of distributing 8 identical balls in 3 distinct boxes so that none of the boxes is empty is-
The different ways in which items from a set may be chosen, usually without replacement, to construct subsets, are called permutations and combinations. When the order of the selection is a consideration, this selection of subsets is referred to as a permutation; when it is not, it is referred to as a combination. So the final answer is 21.
A mass of
strikes the wall with speed
at an angle as shown in figure and it rebounds with the same speed. If the contact time is
, what is the force applied on the mass by the wall
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)
A mass of
strikes the wall with speed
at an angle as shown in figure and it rebounds with the same speed. If the contact time is
, what is the force applied on the mass by the wall
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)
If the locus of the mid points of the chords of the ellipse
, drawn parallel to
is
then ![m subscript 1 end subscript m subscript 2 end subscript equals](data:image/png;base64,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)
If the locus of the mid points of the chords of the ellipse
, drawn parallel to
is
then ![m subscript 1 end subscript m subscript 2 end subscript equals](data:image/png;base64,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)
If nCr denotes the number of combinations of n things taken r at a time, then the expression nCr+1 + nCr –1 + 2 × nCr equals-
The different ways in which items from a set may be chosen, usually without replacement, to construct subsets, are called permutations and combinations. When the order of the selection is a consideration, this selection of subsets is referred to as a permutation; when it is not, it is referred to as a combination. So the final answer is .
If nCr denotes the number of combinations of n things taken r at a time, then the expression nCr+1 + nCr –1 + 2 × nCr equals-
The different ways in which items from a set may be chosen, usually without replacement, to construct subsets, are called permutations and combinations. When the order of the selection is a consideration, this selection of subsets is referred to as a permutation; when it is not, it is referred to as a combination. So the final answer is .