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If the length of a focal chord of the parabola `y^2=4a x`at `a`distance `b`from the vertex is `c ,`then prove that `b^2c=4a^3dot`A. `2a^(2)=bc`B. `a^(3)=b^(2)c`C. `ac=b^(2)`D. `b^(2)c=4a^(3)` |
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Answer» Correct Answer - D Let `P(at_(1)^(2), 2at_(1))" and "Q(at_(2)^(2), 2at_(2))` be the end points of a focal chord of the parabola `y^(2)=4ax`. Then, `PQ=a(t_(2)-t_(1))^(2)` The equation of PQ is `(t_(1)+t_(2))y=2x-2a" "[becauset_(1)ty_(2)=-1]` It is given that PQ=c, and it is at distance b form the vertex. `:." "a(t_(2)-t_(1))^(2)n =c" and, "|(-2a)/(sqrt((t_(1)-t_(2))^(2)+4))|=b` `rArr" "(t_(1)-t_(2))^(2)=c/a" and, "(2a)/((t_(1)-t_(2)))=b" "[becauset_(1)t_(2)=-1]` `rArr" "((2a)/a)^(2)=c/arArr4a^(3)=b^(2)c` |
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