Circles practice · Easy

The locus of the centre of the circle for which one end of a diameter is $\left( {1,1} \right)$ while the other end is on the line $x + y = 3$, is

If the lines $2 x-3 y-5=0$ and $3 x-4 y=7$ are diameters of a circle of area 154 square units, then the equation of the circle is

The equation of the tangents to the circle ${x^2} + {y^2} = 4$, which are parallel to $x + 2y + 3 = 0$, are

The length of the longest ray drawn from the point $(4,3)$ to the circle $x^2+y^2+16 x+18 y+1=0$ is equal to

If the square of the length of the tangent drawn from any point on the circle $x^2+y^2+70 x-45 y+6789=0$ to the circle $x^2+y^2+70 x-45 y+9876=0$ is $K$ then find $K-3080.$

The radius of the circle ${r^2} - 2\sqrt 2 r\left( {\cos \theta + \sin \theta } \right) - 5 = 0,$ is

Find the equation to the circle touching the $y$-axis at a distance -3 from the origin and intercepting a length 8 on the $x$-axis.

$\alpha, \beta$ and $\gamma$ are parametric angles of three points $P, Q$ and $R$ respectively, on the circle $x^2+y^2=1$ and $A$ is the point $(-1,0)$. If the lengths of the chords $A P, A Q$ and $A R$ are in $G P$, then $\cos \left(\frac{\alpha}{2}\right), \cos \left(\frac{\beta}{2}\right)$ and $\cos \left(\frac{\gamma}{2}\right)$ are in

If the points $\left( {2,0} \right),\left( {0,1} \right),\left( {4,5} \right)$ and $\left( {0,c} \right)$ are concyclic, then the value of $c$ is

Let $P Q$ and $R S$ be tangents at the extremities of diameter $P R$ of a circle of radius $r$. If $P S$ and $R Q$ intersect at a point $X$ on the circumference of the circle, then $2 r$ equals

The radius of any circle touching the lines $3x - 4y + 5 = 0$ and $6x - 8y - 9 = 0$ is

Let $S$ be the circle in the $x y$-plane defined by the equation $x^2+y^2=4$. Let $P$ be a point on the circle with both coordinates being positive. Let the tangent to $S$ at $P$ intersect the coordinate axes at $M$ and $N$. Then the mid-point of the line segment MN must lie on the curve

Radius of bigger circle touching the circle $x^2+y^2-4 x-4 y+4=0$ and both the coordinate axes, is

The angle between the tangents drawn at the points $\left( {5,12} \right)$ and $\left( {12, - 5} \right)$ to the circle ${x^2} + {y^2} = 169$ is

The values of $\lambda$ so that the line $3x - 4y = \lambda$ touches ${x^2} + {y^2} - 4x - 8y - 5 = 0$ are

Let the tangent to the circle $C_1: x^2+y^2=2$ at the point $M(-1,1)$ intersect the circle $C_2:(x-3)^2+(y-$ $2)^2=5$, at two distinct points $A$ and $B$. If the tangents to $C_2$ at the points $A$ and $B$ intersect at $N$, then the area of the triangle $A N B$ is equal to

Assertion: Radius of the circle $2 x^2+2 y^2+3 x+4 y+\frac{9}{8}=0$ is 1.

Reason: Radius of the circle $x^2+y^2+2 g x+2 f y+c=0$ is $\sqrt{\left(\frac{1}{2} \text { coeff. of } x\right)^2+\left(\frac{1}{2} \text { coeff. of } y\right)^2-\text { constant term }}$

Let $A(1,\,4)$ and $B\left( {1,\, - 5} \right)$ be two points. Let $P$ be a point on the circle ${(x - 1)^2} + {(y - 1)^2} = 1$, such that ${\left( {PA} \right)^2} + {\left( {PB} \right)^2}$ have maximum value, then the points $P$, $A$ and $B$ lie on

The equation of circle is ${x^2} + {y^2} - 2x = 0$. The point $P\left( { - 1,0} \right)$ lies

If $(a, 0)$ is a point on a diameter of the circle $x^2+y^2=4$, then the equation $x^2-4 x-a^2=0$ has

Equation of the circle with centre on the $y$-axis and passing through the origin and (2,3) is

The locus of the point, such that tangents drawn from it to the circle $x^2+y^2-6 x-8 y=0$ are perpendicular to each other, is

If circles $x^2+y^2+2 a x+c^2=0$ and $x^2+y^2+2 b y+c^2=0$, touch each other then

Tangents $P A$ and $P B$ are drawn to the circle $x^2+y^2=8$ from any arbitrary point $P$ on the line $x+y=4$. The locus of the mid-point of chord of contact $A B$ is

Let the circle $C_1: x^2+y^2-2(x+y)+1=0$ and $C_2$ be a circle having centre at $(-1,0)$ and radius 2. If the line of the common chord of ${C}_1$ and ${C}_2$ intersects the $\mathrm{y}$-axis at the point ${P},$ then the square of the distance of ${P}$ from the centre of ${C}_1$ is:

The equation $\left( {x - {x_1}} \right)\left( {x - {x_2}} \right) + \left( {y - {y_1}} \right)\left( {y - {y_2}} \right) = 0$ represents a circle whose centre is

If $\frac{x}{\alpha}+\frac{y}{\beta}=1$ touches the circle $x^2+y^2=a^2$, then point $\left(\frac{1}{\alpha}, \frac{1}{\beta}\right)$ lies on

If $A=\left\{(x, y): x^2+y^2=25\right\}$ and $B=\left\{(x, y): x^2+9 y^2=144\right\}$ then the number of points, ${A} \cap {B}$ contains is

$\mathrm{A}$ line is drawn through a fixed point $P(\alpha, \beta)$ to cut the circle $x^2+y^2=r^2$ at $A$ and B. Then PA.PB is equal to

The line $x+y \tan \theta=\cos \theta$ touches the circle $x^2+y^2=4$ for