Find the length of the chord of the parabola | vedclass

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(B) Let the vertex be $A(0,0)$. Let $P$ be a point on the parabola $x^2 = 4ay$ such that the chord $AP$ has a slope of $	an \alpha$. The coordinates

Vedclass · Accepted Answer

$4a 	an \alpha \sec \alpha$

Vedclass · Answer

(B) Let the vertex be $A(0,0)$. Let $P$ be a point on the parabola $x^2 = 4ay$ such that the chord $AP$ has a slope of $	an \alpha$. The coordinates of $P$ are $(2at, at^2)$.The slope of $AP$ is given by $\frac{at^2 - 0}{2at - 0} = \frac{t}{2}$.Given the slope is $	an \alpha$,we have $\frac{t}{2} = 	an \alpha$,which implies $t = 2 	an \alpha$.The length of the chord $AP$ is $\sqrt{(2at - 0)^2 + (at^2 - 0)^2} = \sqrt{4a^2t^2 + a^2t^4} = at \sqrt{4 + t^2}$.Substituting $t = 2 	an \alpha$:$AP = a(2 	an \alpha) \sqrt{4 + (2 	an \alpha)^2} = 2a 	an \alpha \sqrt{4(1 + 	an^2 \alpha)}$.Since $1 + 	an^2 \alpha = \sec^2 \alpha$,we get:$AP = 2a 	an \alpha \sqrt{4 \sec^2 \alpha} = 2a 	an \alpha (2 \sec \alpha) = 4a 	an \alpha \sec \alpha$.

Find the length of the chord of the parabola $x^2 = 4ay$ which passes through the vertex and has a slope of $\tan \alpha$.

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