# Pl. answer the following question on parabola

If the locus of middle point of point of contact of tangent drawn to the parabola y^2=8x and foot of perpendicular drawn from its focus to the tangent is a conic then length of latus rectum of this conic is?
asked Jan 5, 2013 in GEOMETRY

+1 vote

Equation of the parabola,y^2=8x

4a=8

a=2

Focus is (2,0)

To graph this parabola, we find the two points that determine the latus rectum by
letting x=2 Then
y^2=8(2)
y^2=16
y=(+ or -)4
The points (2,-4)and (2,4) determine the latus rectum.

Length of the latus rectum is l=8

Length of the latus rectum is 9.

Parabola y² = 8x has focus (2,0)

x = y² / 8

Apply derivative to each side inorder to find out the slope.

dx / dy = 2y / 8 = y / 4.

Tangent line at point (a²/8, a) is x − a²/8 = a/4 (y − a)

x = ay/4 − a²/4 + a²/8

x = ay/4 − a²/8.

Here slope m = a/4, perpendicular slope = -1/m = -4/a.

Line perpendicular to tangent and passing through focus (2,0)

x − 2 = −4/a (y − 0)

x = −4y / a + 2

Foot of perpendicular drawn from focus to tangent is the point of intersection of these two lines:

ay / 4 − a² / 8 = −4y / a + 2

y = a/2

x = 0

So conic is locus of midpoints of (0, a/2) and (a²/8, a) = (a²/16, 3a/4)

y = 3a/4

a = 4y/3

x = a²/16

x = (4y/3)²/16

x = y² / 9

9x = y²

This is equation of parabola with focus = (9/4, 0)

When x = 9/4,  y² = 9(9/4) = 81/4

y = 9/2, −9/2

Therefore length of latus rectum = 9/2 − (−9/2) = 9.