The line $lx + my + n = 0$ is normal to the circle ${x^2} + {y^2} + 2gx + 2fy + c = 0$, if
$lg + mf - n = 0$
$lg + mf + n = 0$
$lg = mf - n = 0$
$lg - mf + n = 0$
A circle is drawn with $y- $ axis as a tangent and its centre at the point which is the reflection of $(3, 4)$ in the line $y = x$. The equation of the circle is
A circle touches the $y$ -axis at the point $(0,4)$ and passes through the point $(2,0) .$ Which of the following lines is not a tangent to this circle?
Tangents $AB$ and $AC$ are drawn from the point $A(0,\,1)$ to the circle ${x^2} + {y^2} - 2x + 4y + 1 = 0$. Equation of the circle through $A, B$ and $C$ is
Let the point $B$ be the reflection of the point $A(2,3)$ with respect to the line $8 x-6 y-23=0$. Let $\Gamma_A$ and $\Gamma_B$ be circles of radii $2$ and $1$ with centres $A$ and $B$ respectively. Let $T$ be a common tangent to the circles $\Gamma_A$ and $\Gamma_B$ such that both the circles are on the same side of $T$. If $C$ is the point of intersection of $T$ and the line passing through $A$ and $B$, then the length of the line segment $AC$ is. . . . . .
A line meets the co-ordinate axes in $A\, \& \,B. \,A$ circle is circumscribed about the triangle $OAB.$ If $d_1\, \& \,d_2$ are the distances of the tangent to the circle at the origin $O$ from the points $A$ and $B$ respectively, the diameter of the circle is :