Circles practice · Mixed difficulty

Equation of a circle which passes through $(3,6)$ and touches the axes, is

The equation of a tangent to the circle $x^2+y^2=25$ passing through $(-2,11)$, is

The length of the tangent drawn to the circle ${x^2} + {y^2} - 2x + 4y - 11 = 0$ from the point $\left( {1,3} \right)$ is

Find the equation of the chord of contact of the tangents drawn from $(1,2)$ to the circle $x^2+y^2-2 x+4 y+7=0$.

The chord of contact of tangents from a point $P$ to a circle passes through $Q$. If $l_1$ and $l_2$ are the lengths of the tangents from $P$ and $Q$ to the circle, then $P Q$ is equal to

The diameter of a circle are along $2x + y - 7 = 0$ and $x + 3y - 11 = 0$. Then, the equation of this circle, which also passes through $\left( {5,7} \right)$, is

The equation of a circle which has a tangent $3 x+4 y=6$ and two normals given by $(x-1)(y-2)=0$ is

Find the equation of the chord of the circle $x^2+y^2+6 x+8 y-$ $11=0$, whose middle point is $(1,-1)$.

If the equation of tangent to the circle ${x^2} + {y^2} - 2x + 6y - 6 = 0$ parallel to $3x - 4y + 7 = 0$ is $3x - 4y + k = 0$, then the value of $k$ are

Three distinct points A, B and C are given in the two-dimensional coordinate plane
such that the ratio of the distance of any one of them from (2, –1) to its distance from
(–1, 5) is 1 : 2. Then the centre of the circle passing through A, B and C is

The shortest distance from the point $(2,-7)$ to the circle $x^2+y^2-14 x-10 y-151=0$ is

the equation $a{x^2} + 2hxy + b{y^2} + 2gx + 2fy + c = 0$ represents a circle, the condition will be

The number of integral points on the circle, touching the parabola $y^2 = 8x$ at (2, 4) and passing through (0, 4), are equal to

The circle whose equations are ${x^2} + {y^2} + {c^2} = 2ax$ and ${x^2} + {y^2} + {c^2} - 2by = 0$ will touch one another externally if

A variable circle passes through the fixed point $\left( {2,0} \right)$ and touches $y$-axis. Then, the locus of its centre is

The equation of normal to the circle $2{x^2} + 2{y^2} - 2x - 5y + 3 = 0$ at $\left( {1,1} \right)$ is

The pole of the straight line $9x + y - 28 = 0$ with respect to the circle $2{x^2} + 2{y^2} - 3x + 5y - 7 = 0$ is

If $f(x+y)=f(x) \cdot f(y)$ for all $x$ and $y, f(1)=2$ and $\alpha_n=f(n), n \in N$, then the equation of the circle having $\left(\alpha_1, \alpha_2\right)$ and $\left(\alpha_3, \alpha_4\right)$ as the ends of its one diameter is

Lines are drawn through the point $P(-2,-3)$ to meet the circle $x^2+y^2-2 x-10 y+1=0$. The length of the line segment $P A, A$ being the point on the circle, where the line meets the circle at coincident points, is

Which of the following statement(s) is true regarding the following circles?

$x^2+y^2-4 x-8 y=0,\, x^2+y^2-8 x-6 y+20=0$

The number of common tangents to circles ${x^2} + {y^2} + 2x + 8y - 23 = 0$ and ${x^2} + {y^2} - 4x - 10y + 9 = 0$, is

The lengths of the intercepts made by any circle on the coordinates axes are equal if the centre lies on the line (s) represented by

If ${O}$ is the origin and $O P, O Q$ are distinct tangents to the circle $x^2+y^2+2 g x+2 f y+c=0$, the circumcenter of the triangle $O P Q$ is

Locus of mid-point of chords of $x^2+y^2+2 g x+2 f y+c=0$ that pass through the origin, is

If $y=m x+c$ is a tangent to the circle $(x-3)^2+y^2=1$ and is also perpendicular to the tangent to the circle $x^2+y^2=1$ at $(1 / \sqrt{2}, / \sqrt{2})$, then