The locus of the intersection point of $x \cos \alpha - y \sin \alpha = a$ and $x \sin \alpha - y \cos \alpha = b$ is

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

$A$ point $P$ moves such that the distance from $(0,2)$ to $P$ is $\frac{1}{\sqrt{2}}$ times the distance of $P$ from $(-1,0)$. Then the locus of the point is

The locus of the centre of the circle which cuts a chord of length $2a$ from the positive $x$-axis and passes through a point on the positive $y$-axis at a distance $b$ from the origin is:

If the lengths of the tangents drawn from $P$ to the circles $x^2+y^2-2x+4y-20=0$ and $x^2+y^2-2x-8y+1=0$ are in the ratio $2:1$,then the locus of $P$ is

The set of all real values of $\lambda$ for which the point $P$ with coordinates $(\lambda, \lambda^2)$ does not lie inside the triangle formed by the lines $x - y = 0$, $x + y - 2 = 0$, and $x + 3 = 0$ is: