The line segment joining the points $(-5, 1)$ and $(3, 2)$ subtends a right angle at a variable point $P$. Find the locus of the point $P$.

The locus of the point which is equidistant from the point $(1,1)$ and the line $x+y+1=0$ is

Let the function $f(x) = 2x^{2} - \log_{e} x$,$x > 0$,be decreasing in $(0, a)$ and increasing in $(a, 4)$. $A$ tangent to the parabola $y^{2} = 4ax$ at a point $P$ on it passes through the point $(8a, 8a - 1)$ but does not pass through the point $(-\frac{1}{a}, 0)$. If the equation of the normal at $P$ is $\frac{x}{\alpha} + \frac{y}{\beta} = 1$,then $\alpha + \beta$ is equal to-

Two perpendicular tangents to the circle $x^2 + y^2 = a^2$ meet at point $P$. The equation of the locus of $P$ is:

The locus of the middle points of chords of the circle $x^2 + y^2 - 2x - 6y - 10 = 0$ which pass through the origin is:

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