If the straight lines $3x - 4y + 4 = 0$ and $6x - 8y - 7 = 0$ are the tangents to the same circle,then the area of that circle (in square units) is

Let $PQ$ and $RS$ be tangents at the endpoints of the diameter $PR$ of a circle of radius $r$. If $PS$ and $RQ$ intersect at a point $X$ on the circumference of the circle,then the length of the chord through $X$ perpendicular to the diameter $PR$ is:

In a $\triangle ABC$,a point $D$ is chosen on $BC$ such that $BD:DC = 2:5$. Let $P$ be a point on the circumcircle of $\triangle ABC$ such that $\angle PDB = \angle BAC$. Then $PD:PC$ is:

If the coordinates of one end of a diameter of the circle $x^2+y^2+4x-8y+5=0$ are $(2,1)$, then the coordinates of the other end are:

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

If $(a, b)$ is the midpoint of the chord $2x - y + 3 = 0$ of the circle $x^2 + y^2 + 6x - 4y + 4 = 0$,then $2a + 3b =$