$A$ straight line passing through the point $(1,0)$ and not parallel to the $x$-axis intersects the curve $2x^2+5y^2-7x=0$ at two points $A$ and $B$. The angle subtended by the line segment $AB$ at the origin is (in $^\circ$)

Tangents are drawn from the point $(17, 7)$ to the circle $x^2 + y^2 = 169$.
Statement-$1$: These tangents are perpendicular to each other.
Statement-$2$: From every point on the circle $x^2 + y^2 = 338$,perpendicular tangents can be drawn to the given circle.

If the line through the point $P(5,3)$ meets the circle $x^2+y^2-2x-4y+\alpha=0$ at $A(4,2)$ and $B(x_1, y_1)$,then $PA \cdot PB$ is equal to

The power of the point $B(-1, 1)$ with respect to the circle $S \equiv x^2+y^2-2x-4y+3=0$ is $p$. If the length of the tangent drawn from $B$ to the circle $S=0$ is $t$,then the point $(2, 3)$ with respect to the circle $S^{\prime}=0$ having centre at $(p, t^2)$ and passing through the origin:

The centre of the circle passing through the point $(0,1)$ and touching the parabola $y=x^{2}$ at the point $(2,4)$ is

If the two circles,$x^2 + y^2 + 2g_1x + 2f_1y = 0$ and $x^2 + y^2 + 2g_2x + 2f_2y = 0$ touch each other,then: