Question

A tangent to the parabola $y^2=4ax$ meets x axis at a point T. This tangent meets the tangent at the vertex at point P. If rectangle TAPQ is completed then the locus of Q is.

• A. Circle
• B. Straight line
• C. parabola
• D. Ellipse

Answer 1

The correct option is C

parabola

Here we will make use of the parametric form for tangents. The tangent at any point B $(a{t}^2,2at)$ to the parabola is,

$ty=x+a{t}^2$

lets draw a figure representing given conditions

Here Q is the required locus point given by (h, k) and forms the vertex of □QPAT. We can obtain points P and T by solving equation of tangents at x and y axis.

$T\equiv (-a{t}^2,0)$

$P\equiv (0,at)$

Also

$A\equiv (0,0)$

Since

$Q\equiv (h,k)$

$h=AT=-a{t}^2$ - - - - - - (1)

k = AP = at - - - - - - (2)

eliminating t from (1) & (2)

$k^2+ah=0$

hence locus of Q is,

$y^2+ax=0$

$y^2=-ax$

which represents a parabola.