The line $ax + by + c = 0$ is normal to the circle $x^2 + y^2 + 2gx + 2fy + d = 0$ if

If $(3,-1)$ is one end of a diameter of the circle $x^2+y^2-2x+4y=0$,then the equation of the tangent at the other end of that diameter is

The line $ax + by + c = 0$ is a normal to the circle $x^2 + y^2 = r^2$. The length of the intercept made by the circle on the line $ax + by + c = 0$ is:

The tangent to the circle $x^2 + y^2 = 5$ at the point $(1, -2)$ also touches the circle $x^2 + y^2 - 8x + 6y + 20 = 0$. Find its point of contact.

If $\theta$ is the angle between the pair of tangents drawn from a point $P(x_1, y_1)$ to the circle $S = x^2 + y^2 + 2gx + 2fy + c = 0$,then which of the following is true?

The point on the curve $x^2+y^2=a^2, y \geq 0$,at which the tangent is parallel to the $x$-axis is