Angle of Intersection of Two Circles

Q. If the circle $x^2+y^2+4x+22y+c=0$ bisects the circumference of the circle $x^2+y^2-2x+8y-d=0(c,d>0),$ then the maximum possible value of $cd$ is

1. $125$
2. $425$
3. $625$
4. $25$

Q. The common chord of the circles $x^2+y^2-4x-4y=0$ and $2x^2+2y^2=32$ subtends at the origin an angle equal to

1. $\frac{\pi }{3}$
2. $\frac{\pi }{4}$
3. $\frac{\pi }{6}$
4. $\frac{\pi }{2}$

Q. The angle between the circles $S_1:x^2+y^2-4x+6y+11=0$ and $S_2:x^2+y^2-2x+8y+13=0$ is

1. ${60}^{\text{o}}$
2. ${45}^{\text{o}}$
3. ${30}^{\text{o}}$
4. ${90}^{\text{o}}$

Q. If the angle of intersection of the circles $x^2+y^2+x+y=0$ and $x^2+y^2+x-y=0$ is $\theta$, then the equation of the line passing through $(1,2)$ and making an angle $\theta$ with the $y$-axis is

1. $x=1$
2. $y=2$
3. $x+y=3$
4. $x-y=3$

Q. If the angle of intersection of the circles $x^2+y^2+x+y=0$ and $x^2+y^2+x-y=0$ is $\theta$, then the equation of the line passing through $(1,2)$ and making an angle $\theta$ with the $y$-axis is

1. $x=1$
2. $y=2$
3. $x+y=3$
4. $x-y=3$

Q. If the circle $x^2+y^2+4x+22y+c=0$ bisects the circumference of the circle $x^2+y^2-2x+8y-d=0(c,d>0),$ then the maximum possible value of $cd$ is

1. $25$
2. $125$
3. $425$
4. $625$

Q. If the circle $C_1:x^2+y^2=16$ intersects another circle $C_2$ of radius $6$ in such a manner that their common chord is of maximum length and has slope $=\frac{1}{2}$. Then the co-ordinates of the centre of the circle(s) $C_2$ is/are

1. $(-2,-4)$
2. $(2,4)$
3. $(-2,4)$
4. $(2,-4)$

Q. The angle between the circles $S_1:x^2+y^2-4x+6y+11=0$ and $S_2:x^2+y^2-2x+8y+13=0$ is

1. ${90}^{\text{o}}$
2. ${60}^{\text{o}}$
3. ${45}^{\text{o}}$
4. ${30}^{\text{o}}$

Q. The equation of the straight line which passes through the point $(2,-3)$ and makes an angle of $\frac{\pi }{4}$ with the axis of $x$ is

1. $x-y=5$
2. $x-y=1$
3. $x+y=5$
4. $xy=1$

Q. Number of real value of $x$ satisfying the equation, $\arctan \sqrt{x(x+1)}+\arcsin \sqrt{x(x+1)+1}=\frac{\pi }{2}$ is

1. 1
2. more than 2
3. 2
4. 0

Q.

If $arg\left(\frac{z-1}{z+1}\right)=\frac{\pi }{4}$, then find the equation of the circle.

Q. If the angle of intersection of the circles $x^2+y^2+x+y=0$ and $x^2+y^2+x-y=0$ is $\theta$, then equation of the line passing through $(1,2)$ and making an angle $\theta$ with the y − axis is

1. $x=1$
2. $y=2$
3. $x+y=3$
4. $x-y=3$

Q.
In in the given figure, $△ABC$ is a right angle isosceles triangle with $AB=BC=12\text{cm}$. An arc is drawn considering $AR$ as the radius, if the area of region $CPQ=BRQ$, then the area of circle with radius $AR$ is
(correct answer + 1, wrong answer - 0.25)

1. $476{\text{cm}}^2$
2. $196\pi {\text{cm}}^2$
3. $576{\text{cm}}^2$
4. $169\pi {\text{cm}}^2$

Q. Prove that the equation to a circle whose radius is $a$ and which touches the axes of coordinates, which are inclined at an angle $\omega$, is
$x^2+2xy\cos \omega +y^2-2a(x+y)\cot \frac{\omega }{2}+a^2{\cot }^2\frac{\omega }{2}=0$.

Q.
In in the given figure, $△ABC$ is a right angle isosceles triangle with $AB=BC=12\text{cm}$. An arc is drawn considering $AR$ as the radius, if the area of region $CPQ=BRQ$, then the area of circle with radius $AR$ is
(correct answer + 1, wrong answer - 0.25)

1. $476{\text{cm}}^2$
2. $196\pi {\text{cm}}^2$
3. $169\pi {\text{cm}}^2$
4. $576{\text{cm}}^2$

Q.

The angle at which the circles $(x-1{)}^2+y^2=10and{x}^2+(y-2{)}^2=5$ intersect is

1. 

2. 

3. 

4. 

Q.

The angle at which the circles $(x-1{)}^2+y^2=10and{x}^2+(y-2{)}^2=5$ intersect is

1. $\frac{\pi }{6}$

2. $\frac{\pi }{4}$

3. $\frac{\pi }{3}$

4. $\frac{\pi }{2}$

Q. The mid point of chord $2x+y-5=0$ of the parabola $y^2=4x$ is

1. $(1,3)$
2. $(2,1)$
3. $(3,-1)$
4. $(\frac{5}{2},0)$

Q. Find the equations of the lines for which $\cot \theta =\frac{1}{2}$, where $\theta$ is the angle of inclination of the line and $y-$ intercept is $\frac{-3}{2}$

Q. If $\theta$ is the acute angle between the lines represented by equation $a{x}^2+2hxy+b{y}^2=0$, then prove that $\tan \theta =\left|\frac{2\sqrt{h^2-ab}}{a+b}\right|,a+b\ne 0$.