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Question:

Let A and B be two distinct points on the parabola y² = 4x. If the axis of the parabola touches a circle of radius r having AB as its diameter, then the slope of the line joining A and B can be

2r

1r

−r

r

Solution:

Let A = (t₁², 2t₁) and B = (t₂², 2t₂)
C = (Centre of circle) = (t₁² + t₂²/2, t₁ + t₂)
Slope of line AB = 2(t₂ − t₁)/(t₂² − t₁²) = 2/(t₁ + t₂)
r = √((t₁ + t₂)²)
Slope of line AB = ±2r