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.

#Algebra

Easy Math Editor

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Comments

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 - 4 years, 9 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$