Prove that a straight line and parabola cannot intersect at more than

1.	Prove that a straight line and parabola cannot intersect at more than two points.
Answer» Let the standard equation of parabola y² = 4ax …..(1) Equation of line be y = mx + c …(2) Solving (1) & (2) (mx + c)² = 4ax ⇒ mx² + 2mcx + c² – 4ax = 0 ⇒ mx² + 2x(mc – 2a) + c² = 0 This equation can not have more than two solutions and hence a line and parabola cannot intersect at more than two points.

Prove that a straight line and parabola cannot intersect at more than two points.

Answer»

Let the standard equation of parabola y² = 4ax …..(1)

Equation of line be y = mx + c …(2)

Solving (1) & (2)

(mx + c)² = 4ax

⇒ mx² + 2mcx + c² – 4ax = 0

⇒ mx² + 2x(mc – 2a) + c² = 0

This equation can not have more than two solutions and hence a line and parabola cannot intersect at more than two points.