Question

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

Answer 1

Answer

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

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)

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