Let A and B be two points on `y^(2)=4ax` such that normals to the curve at A and B meet at point C, on the curve, then chord AB will always pass through a fixed point whose co-ordinates, are

Options

A. (2a, 0)
B. (-a, 0)
C. (-2a, 0)
D. (a, 0)

Answer

B

Explanation

Let `(at_(1)^(2), 2at_(1))" and B"(at_(2)^(2), 2at_(2))` be two points on `y^(2)=4ax` such that the normals at A and B meet at point C on the curve.
`:." "t_(1)t_(2)=2.`
The equation of the chord AB is
`y(t_(1)+t_(2))=2x+2at_(1)t_(2)`
`rArr" "(2x+4a)-(t_(1)+t_(2))y=0" "[becauset_(1)t_(2)=2]`
Clearly, it passes through the intersection of the lines
`2x+2a=0 " and " y = 0 " i.e. (-a, 0).`