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 midpoint of QR is
(2, 4)
For the parabola y2=8x, the point P(2, 4) corresponds to t =1.
slope of targent at at2,2at is 1t
Hence the tangent to the parabola y2=8x at (2, 4) is
y−4=11(x−2)⇒x−y+2=0⇒y=x+2y=x+2 intersects the parabola y2=8x+5⇒(x+2)2=8x+5⇒x2+4x+4=8x+5⇒x2−4x−1=0Solving for x,x=4 ± √16−42=4 ± 2√32=2±√3
corresponding y coordinates are
x+2⇒4±√3⇒Q=(2+√3,4+√3)⇒R=(2−√3,4−√3)
The midpoint of QR =(2, 4)