If the line $3x - 4y = \lambda$ touches the circle $x^2 + y^2 - 4x - 8y - 5 = 0$,then $\lambda$ is equal to

If a line passing through the origin touches the circle $(x - 4)^2 + (y + 5)^2 = 25$,then its slope is:

Let a circle $C$ of radius $5$ lie below the $x$-axis. The line $L_{1}: 4x + 3y + 2 = 0$ passes through the centre $P$ of the circle $C$ and intersects the line $L_{2}: 3x - 4y - 11 = 0$ at $Q$. The line $L_{2}$ touches $C$ at the point $Q$. Then the distance of $P$ from the line $5x - 12y + 51 = 0$ is

If $2x - 3y + 5 = 0$ and $4x - 5y + 7 = 0$ are the equations of the normals drawn to a circle and $(2, 5)$ is a point on the given circle,then the radius of the circle is

The circle $x=5 \cos \theta, y=5 \sin \theta$ is bounded by the rectangle formed by the lines $x \pm 6=0$ and $y \pm 6=0$. The area of the triangle that lies inside the rectangle,which is formed by the tangent at $P\left(\frac{2 \pi}{3}\right)$ to the circle with two of the above given lines,is

What is the equation of the tangent to the curve $x^2(x - y) + a^2(x + y) = 0$ at the origin?