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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, 2 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 \(C\)

Neelesh Vij - 1 year, 2 months ago

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