If the tangent at the point `P(2,4)` to the parabola `y^2=8x` meets the parabola `y^2=8x+5` at `Qa n dR ,` then find the midpoint of chord `Q Rdot`

Explanation

The equation of the tangent to parabola `y^(2)=8x` at P(2,4) is
4y=4(x+2)
or x-y+2=0 (1)
This chord meets the parabola `y^(2)=8x+5` at Q and R.
Let `(x_(1),y_(1))` be the midpoint of chord QR.n
Then equation of QR is
`yy_(1)-4(x+x_(1))-5=y_(1)^(2)-8x_(1)-5" ""(Using "T=S_(1))`
`or" "4x-yy_(1)-4x_(1)+y_(1)^(2)=0` (2)
Clearly, (1) and (2) represent the same line.
`So," "(4)/(1)=-(y_(1))/(-1)=(-4x_(1)+y_(1)^(2))/(2)`
`rArr" "y_(1)=4and8=-4x_(1)+y_(1)^(2)`
`rArr" "y_(1)=4andx_(1)=2`