The equations of the tangents drawn from the point $(0, 1)$ to the circle ${x^2} + {y^2} - 2x + 4y = 0$ are
$2x - y + 1 = 0,\,\,x + 2y - 2 = 0$
$2x - y + 1 = 0,\,\,x + 2y + 2 = 0$
$2x - y - 1 = 0,\,\,x + 2y - 2 = 0$
$2x - y - 1 = 0,\,\,x + 2y + 2 = 0$
From the origin chords are drawn to the circle ${(x - 1)^2} + {y^2} = 1$. The equation of the locus of the middle points of these chords is
The line $x\cos \alpha + y\sin \alpha = p$will be a tangent to the circle ${x^2} + {y^2} - 2ax\cos \alpha - 2ay\sin \alpha = 0$, if $p = $
Let the normals at all the points on a given curve pass through a fixed point $(a, b) .$ If the curve passes through $(3,-3)$ and $(4,-2 \sqrt{2}),$ and given that $a-2 \sqrt{2} b=3,$ then $\left(a^{2}+b^{2}+a b\right)$ is equal to ..... .
The equations of the tangents to the circle ${x^2} + {y^2} = {a^2}$ parallel to the line $\sqrt 3 x + y + 3 = 0$ are
If $\theta $ is the angle subtended at $P({x_1},{y_1})$ by the circle $S \equiv {x^2} + {y^2} + 2gx + 2fy + c = 0$, then