Consider the parabola y² = 8x. Let Δ₁ be the area of the triangle formed by the end points of its latus rectum and the point P(1/2, 2) on the parabola, and Δ₂ be the area of the triangle formed by drawing tangents at P and at the end points of the latus rectum. Then Δ₁/Δ₂ is?

Given equation of the parabola is y² = 8x ∴ a = 2
The endpoints of its latus rectum are (a, 2a) and (a, -2a)
Points are (2, 4) and (2, -4)
Area of the triangle Δ₁ = 0.5 |2 1/2 4
1 2 -4
1| = 6
The equation of the tangent at (2, 4) is given by y = x + 2…(1)
equation of the tangent at (2, -4) is given by -y = x + 2…(2)
and the equation of the tangent at (0.5, 2) is given by y = 2x + 1…(3)
Intersection points of (1), (2) and (3) are : (-2, 0), (-1, -1) and (1, 3)
Δ₂ = | -2 1 1
-1 -1 1
1 3 1| = 3
∴ Δ₁/Δ₂ = 2