Question

If a circle intersects the parabola y² = 4ax in four points then show that the algebraic sum of the ordinates is zero. Also show that the line joining one pair of these four points and the line joining the other pair have slopes equal in magnitude. 

Answer 1

Solving y² = 4ax and x² + y² + 2gx + 2fy + c = 0

we have y₁ + y₂ + y₃ + y₄ = 0 (y₁ , y₂ , y₃ , y₄ are ordinates of the points of intersection)

y₁ + y₂= – (y₃+ y₄)

Slope of line joining (x₁ , y₁) and (x₂ , y₂) is m (say)

Slope of line joining (x₃ y₃) and (x₄ , y₄) is m' (say)

Slope of line joining (x₃ y₃) and (x₄ , y₄) is m' (say)

Since y₁ + y₂ = –(y₃ + y₄),

We have m = – m'

⇒ Slopes are equal in magnitude.