Question

If a tangent to the parabola $y^2=4ax$ meets the x-axis at T and the tangent at the vertex A, at P. Let the rectangle TAPQ be completed. Then the locus of the point Q is

• A. a circle
• B. a straight line
• C. a parabola
• D. an ellipse

Answer 1

The correct option is C

a parabola

The tangent at any point $B(a{t}^2,2at)$ to the parabola is $ty=x+a{t}^2.......\left(1\right)$

Since tangent at the vertex A is the y-axis, T and P are $(-a{t}^2,0)and(0,at)$ respectively. Clearly A is (0, 0)
If Q be(h, k), then h =AT =$-a{t}^2$ and k= AP = at Eliminating t, we get $k^2+ah=0$
Hence the locus of Q is $y^2+ax=0,$ which is parabola.