Common Chord of Two Circles

If the circle  $C_1:x^2+y^2=16$ intersects another circle $C_2$ of radius $5$ in such a manner that the common chord is of maximum length and has a slope equal to  $\frac{3}{4}$ , then the coordinates of the centre of $C_2$ are

Equation of the straight line meeting the circle with centre at origin and radius equal to $5$ in two points at equal distances of $3$ units from the point $A\left(3, 4\right)$ is

The equation of the common chord of the pair of circles: $(x-a{)}^2+(y-b{)}^2=c^2,$ and $(x-b{)}^2+(y-a{)}^2=c^2(a\ne b)$ is

Two circles of radii $6$ and $8$ units intersect orthogonally. If the length of common chord is $\lambda$, then $5\lambda$ is

The circle $x^2+y^2+2gx+2fy+c=0$ bisects the circumference of the circle $x^2+y^2+2g_{1}x+2f_{1}x+c_1=0$. Prove that $2g_1\left(g-g_1\right)+2f_1\left(f-f_1\right)=c-c_1$.

Prove that the equation of the circle of which a diameter is the common chord of two circles $x^2+y^2-4x-8=0$ and $x^2+y^2+8x+4=0$ is $x^2+y^2+2x-2=0$.

Tangents are drawn to the circle $x^2+y^2=1$ at the points where it is met by the circles $x^2+y^2-(\mu +6)x+(8-2\mu )y-3=0$ where $\mu$ being variable, the locus of point of intersection of these tangents is

If the circle $x^2+y^2+4x+22y+c=0$ bisects the circumference of the circle $x^2+y^2-2x+8y-d=0,$ then find $c+d$.

If the circumference of the circle $x^2+y^2+8x+8y-b=0$ is bisected by the circle $x^2+y^2-2x+4y+a=0$ then find $\left|a+b\right|$.

The circles having $r_1$ and $r_2$ intersect orthogonally. Then, length of their common chord is

The locus of centre of circles which bisect the circumference of circles $x^2+y^2=9$ and $x^2+y^2-2x-4y+2=0$ is

Two circles whose radii are equal to $4$ and $8$ intersects at right angles. The length of their common chord is

The possible number of values of $a$ for which the common chord of the circles $x^2+y^2=8$ and ${\left(x-a\right)}^2+y^2=8$ subtends a right angle at the origin is

Let $\mathrm{P}\mathrm{Q}$ be the common chord of the circles $S_1:x^2+y^2+2x+3y+1=0$ and $S_2:x^2+y^2+4x+3y+2=0$, then the perimeter (in units) of the triangle ${\mathrm{C}}_1\mathrm{P}\mathrm{Q}$ is equal to $\left(\mathrm{where},\hspace{0.17em}{C}_1=\left(-1,\frac{-3}{2}\right)\right)$

Tangents are drawn to the circle $x^2+y^2=12$ at the points where it is met by the circle $x^2+y^2-5x+3y-2=0,$ then which one is the $x-$ coordinate of the point of intersection of these tangents

Two tangents are drawn to the circle $x^2+y^2=16$ to the intersection point of circle $x^2+y^2=16$ with other circle having equation$x^2+y^2-6x-8y-8=0,$ then the point of intersection of their tangents is

If the circle $x^2+y^2+4x+22y+c=0$ bisects the circumference of the circle $x^2+y^2-2x+8y-d=0$ then $(\mathrm{c}+\mathrm{d})$ is

For $\mathrm{\Delta ABC}, \mathrm{AB}=4, \mathrm{AC}=2$ and $\angle \mathrm{A}=\frac{2\pi }{3}$ Circles are described with $\mathrm{AB}$ and $\mathrm{AC}$ as diameters. A circle of largest possible radius $\text{'}\mathrm{r}\text{'}$ is inscribed on the common part of above two circles. If $2\mathrm{r}=\sqrt{\mathrm{m}}-\sqrt{\mathrm{n}}$ where $\mathrm{m},\mathrm{n}\in \mathrm{N}$ ,then tick the appropriate optionn

If the circle ${\omega }_1:x^2+y^2+4x+22y+c=0$ bisects the circumference of the circle ${\omega }_2:x^2+y^2-2x+8y-d=0$, find $c+d$.

If the line $x\cos \alpha +y\sin \alpha =p,$ represents the common chord $\mathrm{APQB}$ of the circles $x^2+y^2=a^2$ and $x^2+y^2=b^2(a>b)$ as shown in the diagram, then $\mathrm{PA}$ is equal to _____