# Parabolic Proof

Prove the following:

Given a fixed point $$P(c,d)$$ inside parabola $$x^2 = 4ay$$, where $$a$$ is the focal length, we draw a chord through $$P$$ intersecting the parabola at $$A$$ and $$B$$. Prove the locus of the intersections of the tangents to the parabola at $$A$$ and $$B$$ is the line $$y=\dfrac {c}{2a} - d$$. (The chord can be variable)

Extension: Prove the general form.

Note by Sharky Kesa
1 year, 10 months ago

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Hint: Let the chord be $$XY$$ then this chord will be the chord of contact for the point $$C (x_1,y_1)$$ which is point of intersection of tangents.

Equation of a chord of contact can be easily derived. here it comes out to be $$xx_1 -2a(y+y_1) =0$$

Now it passes through $$P$$

So plug it in the equation then you get the required locus for the point $$C$$

- 1 year, 10 months ago