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Let `A(x_(1), y_(1))" and "B(x_(2), y_(2))` be two points on the parabola `y^(2)=4ax`. If the circle with chord AB as a diameter touches the parabola, then `|y_(1)-y_(2)|=`A. 4aB. 8aC. `6sqrt2a`D. none of these |
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Answer» Correct Answer - B Let the coordinates of A and B `(at_(1)^(2), 2at_(2))" and " (at_(2)^(2), 2at_(2))` respectively. Then, equation of the circle described on AB as a diameter is `(x-at_(1)^(2))(x-at_(2)^(2))+(y-2at_(1))(y-2at_(2))=0` Suppose this cuts `y^(2)-4ax" at "(at^(2), 2at)`. Then, `(at^(2)-at_(1)^(2))(at^(2)-at_(2)^(2))+(2at+2at_(1))(2at-2at_(2))=0` `rArr" "(t+t_(1))(t+t_(2))+4=0` `rArr" "t^(2)+t(t_(1)+t_(2))+t_(1)t_(2)+4=0` If the circle touches the parabola, then this equation must give equal values of t. `:." "(t_(1)+t_(2))^(2)-4(t_(1)t_(2)+4)=0` `rArr" "(t_(1)-t_(2))^(2)=16rArr|t_(1)-t_(2)|=4rArr|y_(1)-y_(2)|=8a` |
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