The triangle $PQR$ is inscribed in the circle $x^2 + y^2 = 25$. If $Q$ and $R$ have coordinates $(3, 4)$ and $(-4, 3)$ respectively,then $\angle QPR$ is equal to

The point $(5, -7)$ lies outside the circle:

If $(a \cos \theta_i, a \sin \theta_i)$ for $i = 1, 2, 3$ represent the vertices of an equilateral triangle inscribed in a circle $x^2 + y^2 = a^2$,then:

Let $G$ be a circle of radius $R>0$. Let $G_1, G_2, \ldots, G_n$ be $n$ circles of equal radius $r>0$. Suppose each of the $n$ circles $G_1, G_2, \ldots, G_n$ touches the circle $G$ externally. Also,for $i=1,2, \ldots, n-1$,the circle $G_i$ touches $G_{i+1}$ externally,and $G_n$ touches $G_1$ externally. Then,which of the following statements is/are $TRUE$?
$(A)$ If $n=4$,then $(\sqrt{2}-1)r < R$
$(B)$ If $n=5$,then $r < R$
$(C)$ If $n=8$,then $(\sqrt{2}-1)r < R$
$(D)$ If $n=12$,then $\sqrt{2}(\sqrt{3}+1)r > R$

If the ratio of the lengths of the tangents drawn from the point $(1, 2)$ to the circles $x^2 + y^2 + x + y - 4 = 0$ and $3x^2 + 3y^2 - x - y + k = 0$ is $4 : 3$,then $k = \dots$

The circle $x^2 + y^2 = 4x + 8y + 5$ intersects the line $3x - 4y = m$ at two distinct points if: