Circle

A stick of length $r$ units slides with its ends on coordinate axes. Then, the locus of the mid-point of the stick is a curve whose length is

Find the equation of the circle which cutting orthogonally to the circle $x^2+y^2-4x+2y-7=0$ having centre $(2,3).$

The acute angle between the line $3x-4y=5$ and the circle $x^2+y^2-4x+2y-4=0$ is $\theta ,$ then find the value of $\sin \theta .$

What is radical axis $?$

What is the director circle$?$

The line $x+y=2\sqrt{2}$ touches the circle $x^2+y^2=4$ at:

The four distinct points $(0,0),(2,0),(0,-2) \text{and} (k,-2)$ are con-cyclic if k=

The abscissae and ordinates of the endpoints A and B of a focal chord of the parabola $y^2=4x$ are respectively, the roots of equations $x^2-3x+a=0$and $y^2+6y+b=0$. The find the equation of the circle with AB as diameter.

The length of the chord cut by $4y=3x+1$ on the circle $x^2+y^2-6x=0$

Find the points on the circle $x^2+y^2-2x+4y-20=0$ which are the farthest and nearest to the point $(-5,6)$. 

Find the radius of the circle $a{x}^2+b{y}^2+(a+b-4)xy-ax-by-20=0$

Prove that the centre of three circles $x^2+y^2+2x+6y+1=0, x^2+y^2-12y-9=0, x^2+y^2=25$ lies on the same straight line. What is the equation of this line?

Find the equation of a circle passing through $\left(2,-2\right)$ and $\left(3,4\right)$ whose centre lies on the line $x+y=2$.

Find coordinates of centre of a circle and its radius, $x^2+y^2-6x+8y=0$

$1.$Two given circles $x^2+y^2+ax+by+c=0 \& x^2+y^2+dx+ey+f=0$ will intersect each other orthogonally when

   $$\begin{gathered} a. a+b+c=d+e+f b. ad+be=c+f \\ c. ad+be=2c+2f d. 2ad+2be=c+f \end{gathered}$$

$2.$If the angle between the circles $x^2+y^2-4x-6y+k=0 \& x^2+y^2+8x-4y+11=0 \text{is} \frac{\mathrm{\pi }}{3}, \text{then the value of} k \text{is}$

$a. -3 b. 36 c.3 d.2$

$3.$The angle at which the circles $x^2+y^2+8x-2y-9=0, x^2+y^2-2x+8y-7=0 \text{intersect is}$

$a. \text{obtuse} b. \frac{\mathrm{\pi }}{6} c. \frac{\mathrm{\pi }}{3} d. \frac{\mathrm{\pi }}{2}$

$4.$The angle between the circles $x^2+y^2=a^2, x^2+y^2=ax+ay \text{is }$

$a. \frac{\mathrm{\pi }}{4} b. \frac{3\mathrm{\pi }}{4} c. \frac{\mathrm{\pi }}{2} d. \frac{\mathrm{\pi }}{3}$

$5.$ Find $k,$ if the pair of circles $x^2+y^2-5x-14y-34=0, x^2+y^2+2x+4y+k=0 \text{are othogonal}$

$a. -1 b. 1 c. 2 d. -2$

The circle ${\left(x-h\right)}^2+{\left(y-k\right)}^2=r^2$ touches $x-$axis at $\left(1,0\right).$ Also the chord $x+y=0$ of the circle has length $\text{'}2\text{'}$, then $h+k+r=?$

Find the equation of the circle which passes through the points $\left(2, -2\right)$, and $\left(3, 4\right)$ and whose centre lies on the line $x+y=2$.

Find the centre and the radius of the circle $x^2+y^2+8x+10y-8=0$

Find the equation of the circle with centre $\left(-3, 2\right)$ and radius $4$.

Find an equation of the circle with centre at $\left(0, 0\right)$ and radius $r$.