If a point $P(x, y)$ moves such that the sum of the squares of its coordinates is equal to their product,then the locus of $P$ excluding the origin is

The equation of the locus of a point whose distance from $(a, 0)$ is equal to its distance from the $y$-axis is

The side $AB$ of $\triangle ABC$ is fixed and is of length $2a$ units. The vertex $C$ moves in the plane such that the vertical angle $\angle ACB$ is always constant and is equal to $\alpha$. Let the $x$-axis be along $AB$ and the origin be at $A$. Then the locus of the vertex $C$ is:

The locus of the centre of the circle $\frac{1}{2} (x^2 + y^2) + x \cos \theta + y \sin \theta - 4 = 0$ is :-

If the equation of a curve remains unchanged by replacing $x$ and $y$ with $-x$ and $-y$ respectively,then the curve is

The locus of the points from which perpendicular tangents can be drawn to the circle $x^2 + y^2 = a^2$ is