Let $AB$ be the diameter of a semicircle $S$. The locus of the centres of circles which are tangent to $AB$ and to $S$ is an arc of

For any value of $\theta$,if the straight lines $x \sin \theta + (1 - \cos \theta) y = a \sin \theta$ and $x \sin \theta - (1 + \cos \theta) y + a \sin \theta = 0$ intersect at $P(\theta)$,then the locus of $P(\theta)$ is a

The locus of the points on the curve ${y^2 = 4a(x + a \sin \frac{x}{a})}$ where the tangent is parallel to the $x$-axis represents . . . .

If a circle passes through the point $(a, b)$ and cuts the circle $x^2 + y^2 = K^2$ orthogonally,then the equation of the locus of its centre is:

The point of concurrence of all the chords of the curve $3x^2 - y^2 - 2x + 4y = 0$ which subtend a right angle at the origin is

If $\frac{x}{\alpha} + \frac{y}{\beta} = 1$ touches the circle $x^2 + y^2 = a^2$,then the point $(\frac{1}{\alpha}, \frac{1}{\beta})$ lies on a/an