Question

\((1,1)\) is the focus of the parabola \[ y^{2}-4ax-2ay+a^{2}=0. \] If the circles \[ (x-\alpha)^2+(y-\beta)^2=r^2 \] touch the X-axis and the axis of the given parabola, then \[ \{(\alpha,\beta)\} \] is:

• A. a line \(y=\frac12\)
• B. a line \(y=1\)
• C. a circle \(x^2+y^2=\frac14\)
• D. a parabola \(y^2=2x\)

Hint:

For a circle tangent to two lines, equate the perpendicular distances from the centre to both lines.

Answer 1

The Correct Option is A