The length of the chord of the parabola x^2 = | vedclass

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(B) Let the vertex of the parabola $x^2 = 4ay$ be $A(0, 0)$.Let the other end of the chord be $P(x_1, y_1) = (2at, at^2)$.The slope of the chord $AP$

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$4a 	an\alpha \sec\alpha$

Vedclass · Answer

(B) Let the vertex of the parabola $x^2 = 4ay$ be $A(0, 0)$.Let the other end of the chord be $P(x_1, y_1) = (2at, at^2)$.The slope of the chord $AP$ is given by $m = \frac{at^2 - 0}{2at - 0} = \frac{t}{2}$.Given that 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 $t = 2	an\alpha$:$AP = a(2	an\alpha)\sqrt{4 + (2	an\alpha)^2} = 2a	an\alpha \sqrt{4(1 + 	an^2\alpha)} = 2a	an\alpha \sqrt{4\sec^2\alpha} = 2a	an\alpha (2\sec\alpha) = 4a	an\alpha \sec\alpha$.

The length of the chord of the parabola $x^2 = 4ay$ passing through its vertex and having slope $\tan\alpha$ is:

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