Question

The line 2x – y + 4 = 0 cuts the parabola y² = 8x in P and Q. The mid - point of PQ is :

A. (1, 2) 

B. (1, - 2) 

C. (- 1, 2) 

D. (- 1, - 2)

Answer 1

Option : (C)

Given that,

The line 2x - y + 4 = 0 cuts the parabola y² = 8x at P, Q. 

We need to find the midpoint of PQ. 

Substituting y = 2x + 4 in the equation of parabola. 

⇒ (2x + 4)² = 8x 

⇒ 4x² + 16x + 16 = 8x 

⇒ 4x² + 8x + 16 = 0 

⇒ x² + 2x + 4 = 0 

Let x₁, x₂ be the roots. 

Then, 

x₁ + x₂ = - 2 

Now,

Substituting x = \(\frac{y^2}{8}\)in the equation of the line we get,

⇒ y² - 4y + 16 = 0 

Let y₁, y₂ be the roots. 

Then, 

y₁ + y₂ = 4 

Let us assume P be (x₁, y₁) and Q be (x₂, y₂) and R be the midpoint of PQ.

⇒ R = (- 1, 2)