A set of parallel chords of the parabola `y^2=4a x` have their midpoint on (a)any straight line through the vertex (b)any straight line through the focus (c)a straight line parallel to the axis (d)another parabola

Options

A. any straight line through the vertex
B. any straight line through the focus
C. a straight line parallel to the axis
D. another parabola

Answer

C

Explanation

(3) Let point `P(at_(1)^(2),2at_(1))andQ(at_(2)^(2),2at_(2))` lie on the parabola `y^(2)=4ax`.
Here, point P and Q are variable. But the slope of chord PQ,
`m_(PQ)=(2)/(t_(1)+t_(2))`
Now, let the midpoint of PQ R(h,k). Then ,
`k=(2at_(1)+2at_(2))/(2)`
`or" "k=a(t_(1)+t_(2))=(2)/(m)`
`:." "y=(2)/(m)`
which is a line parallel to the axis of the parabola.