Question

obtain the condition that the line will touch the parabola
$y {2} = 4ax$

$lx - my + n = 0$
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Answer 1

Answer:

Let us solve the two equations to obtain the point of intersection.

y2=4axy2=4ax

⟹x=y24a(1)(1)⟹x=y24a

lx+my+n=0(2)(2)lx+my+n=0

Substitute (1)(1) in (2)(2) to get:

(l4a)y2+my+n=0(3)(3)(l4a)y2+my+n=0

The line, lx+my+n=0lx+my+n=0, touches the parabola y2=4axy2=4ax at only one point, i.e., the line is tangent to the parabola. This means, that when we solve for the point of intersection of parabola and the line, we must get only one solution. For this to happen, the discriminant of the (3)(3) must be zero. Therefore,

m2−4×l4a×n=0m2−4×l4a×n=0

⟹m2−lna=0⟹m2−lna=0

⟹am2=ln⟹am2=ln

⟹a=(−lm)(−nm)(4)(4)⟹a=(−lm)(−nm)

Notice that the slope of the line (M)(M) is −lm−lm and the y-intercept of the line (C)(C) is −nm−nm. Using these two facts, (4)(4) can be written as:

a=MCa=MC

To sum up, a line is said to be a tangent to a parabola if the product of the slope and y-intercept of the line is equal to the focal length of the parabola.