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Question

If the tangent at the point P(2,4) to the parabola y2=8x meets the parabola y2=8x+5 at Q and R, then the sum of the coordinates of the mid point of QR is ​​

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Solution

Equation of tangent at point P(2,4) to parabola y2=8x is
4y=4(x+2)4x4y+8=0(1)

Let (x1,y1) be the mid point of the chord QR
Equaion of chord of the parabola y2=8x+5 whose mid point is (x1,y1) will be
T=S1yy14(x+x1)5=y218x154xyy1+y214x1=0(2)

Both equation (1) and (2) represents the same equation
Hence, comparing (1) and (2), we get
44=y14=y214x18
y1=4
x1=2

The required sum is 4+2=6

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