Question

A circle drawn through any point $P$ on the parabola $y^2=4x$ has its centre on the tangent drawn at $P$. The circle also passes through the point of intersection of tangent and directrix $T$. Then the circle passes through

• A. $(1,0)$
• B. $(0,0)$
• C. $(\frac{1}{2},0)$
• D. $(-1,0)$

Answer 1

The correct option is A $(1,0)$

The circle passes through point P and point T in the diagram shown.
Also, its centre will be at some point on the tangent through P and so we have the end points of diameter as point P and tangent's intersection with the directrix, point T.

In parametric form the coordinates of point 'P' can be taken as $(t^2,2t)$

Hence, equation of tangent at 'P' is $y\times 2t=2(x+t^2)$

The equation of directrix is $x=-1$.

So, the coordinates of point of intersection of tangent and directrix are$T:(-1,t-\frac{1}{t})$
Since, 'P' and 'T' are the end points of diameter of the circle, the equation of circle can be written as $(x+1)(x-t^2)+(y-2t)(y-t+\frac{1}{t})=0$

$(1,0)$ satisfies the above equation.
$\Rightarrow$ the circle passes through the point $(1,0)$