The tangent at the point `P(x_1,y_1)` to the parabola `y^2=4ax` meets the parabola `y^2=4a(x+b)` at Q and R, the coordinates of the mid-point of QR are :

Options

A. `(x_(1), y_(1))`
B. `(x_(1)+b, y_(1))`
C. `(x_(1)+b,y_(1)+b)`
D. `(x_(1)-b, y_(1)-b)`

Answer

A

Explanation

The equation of the tangent to `y^(2)=8x" at "P(2, 4)` is
`4y=4(x+2)rArrx-y+2=0" …(i)"`
Let `(x_(1), y_(1))` be the mid-point of chord QR. Then, equation of QR is
`yy_(1)-4(x+x_(1))-5=y_(1)^(2)-x_(1)-5`
`rArr" "4x-yy_(1)-4x_(1)+y_(1)^(2)=0" ...(ii)"`
Clearly, (i) and (ii) represent the same line.
`:." "1/1=-y_(1)/(k)=(2ax_(1))/(k^(2)--2ah)`
`rArr" "k=y_(1)" and "k^(2)-2ah=2ax_(1)`
`rArr" "k=y_(1)" and "y_(1)^(2)-2ah=2ax_(1)`
`rArr" "l=y_(1)" and "4ax_(1)-2ah=2ax_(1)[because(x_(1),y_(1))"lies on "underset(thereforey_(1)^(2)=4ax_(1))(y^(2)=4ax)]`
`rArr" "k=y_(1)" and "h=x_(1)`.
Hence, the mid-point of QR is `(x_(1), y_(1))`.