Question

Tangents drawn from the point (
−8, 0) to
the parabola
y2=
8
x touch the parabola at
P and Q. If F is the focus of the parabola,
then the area of the triangle PFQ
(in sq. units) is equal to :
(1) 24
(2) 32
(3) 48
(4) 64

Answer 1

equation of parabola is y² = 8x
differentiate with respect to x,
dy/dx = 4/y
parametric equation of parabola is (at², 2at)
so, (2t², 4t) is parabola equation of y² = 8x

at (2t², 4t) , slope of parabola , dy/dx = 4/4t = 1/t

again, slope of line joining (-8, 0) and (2t², 4t)
= (4t - 0)/(2t² + 8)
= 2t/(t² + 4)

hence, slope of tangent of parabola= slope of line joining (-8, 0) and (2t², 4t)

1/t = 2t/(t² + 4)

t² + 4 = 2t² => t = ±2

so, points are (8, -8) and (8, 8)

now, triangle formed by Focus, F (2, 0) and Points on parabola P(8, 8) and (8, -8)

area of triangle = 1/2 [ 2(8 + 8) + 8(-8-0) + 8(0-8)]
= 1/2 [ 32 - 64 - 64 ] = 48 sq unit.

Answer 2

hey mate,
your answer is in the attachment