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The radius of the circle whose centre is (-4,0) and which cuts the parabola `y^(2)=8x` at A and B such that the common chord AB subtends a right angle at the vertex of the parabola is equal toA. 4B. 3C. `sqrt(18)`D. 5 |
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Answer» Correct Answer - A Let be the radius of the circle. Then, its equation is `(x+)^(2)+y^(2)=r^(2)" ...(i)"` This cute the parabola `y^(2)=8x` at points `A(x_(1), y_(1))` and `B(x_(2), y_(2))`. The abcissae of A and B are the roots of the equation `(x+4)^(2)+8x=r^(2)or, x^(2)+16x+16-r^(2)=0` `:." "x_(1)x_(2)=16-r^(2)" ....(ii)"` The ordinates of A and B are given by `y_(1)^(2)=8x_(1)" and "y_(2)^(2)=8x_(2)` respectively. `:." "y_(1)=+-2sqrt2x_(1)" and "y_(2)=+-2sqrt2x_(2)` Since AB subtends a right angle at the vertex of the parabola. `:." "y_(1)/x_(1)xxy_(2)/x_(2)=-1` `rArr" "x_(1)x_(2)+y_(1)y_(2)=0` `rArr" :x_(1)x_(2)+8x_(1)x_(2)=0` `rArr" "x_(1)x_(2)=0" "["Using (i)"]` `rArr" "16-r^(2)=0rArrr=4` |
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