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If a focal chord of `y^2=4a x`makes an angle `alpha in [0,pi/4]`with the positive direction of the x-axis, then find the minimum length ofthis focal chord.A. `4a sec^(2)alpha`B. `2a "cosec"^(2)alpha`C. `4a" cosec"^(2)alpha`D. `4a cot^(2)alpha` |
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Answer» Correct Answer - C Let `P(at_(1)^(2), 2at_(1))" and Q"(at_(2)^(2), 2at_(2))` be the end points of a focal chord PQ which makes an angle `alpha` with the axis of the parabola. Then, `PQ=a(t_(2)-t_(1))^(2)` Also, `tan alpha" = slope of PQ"` `rArr" "tan alpha=2/(t_(2)+t_(1))rArrt_(1)+t_(2)=2 cot alpha` `:." "PQ=a(t_(2)-t_(1))^(2)` `rArr" "PQ=a{(t_(2)+t_(1))^(2)-4t_(1)t_(2)}` `rArr" "PQ=a {4 cot^(2)alpha+4}" "[becauset_(2)+t_(1)=2 cot alphaandt_(1)t_(2)=-1}` `rArr" "PQ=4a" cosec"^(2)alpha.` |
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