Maths-
General
Easy
Question
If the tangent at the point P(x1, y1) to the parabola y2 = 4ax meets the parabola y2 = 4a(x + b) at Q and R, then the mid-point of QR is -
- (x1 + b, y1 + b)
- (x1 – b, y1 – b)
- (x1, y1)
- (x1 + b, y1)
The correct answer is: (x1, y1)
Equation of the tangent at P(x1, y1) to y2 = 4ax is
yy1 – 2ax – 2ax1 = 0....(1)
Equation of the chord of y2 = 4a(x + b) whose mid-point is (x', y') is
yy' – 2ax – 2ax' – 4ab = y'2 – 4ax' – 4ab
(i.e.) yy' – 2ax – (y'2 – 2ax') = 0
Equations (1) and (2) represent the same line
This gives y' = y1 and then 2ax1 = y'2 – 2ax'
= y12 – 2ax'
= 4ax1 – 2ax'
x' = x1
mid-point (x', y') = (x1, y1).
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