Orthogonality of Two Circles

Q. The equation of the circle which cuts orthogonally each of the three circles given below:
$x^2+y^2-2x+3y-7=0$, $x^2+y^2+5x-5y+9=0$ and $x^2+y^2+7x-9y+29=0$

1. $x^2+y^2-16x-8y-12=0$
2. $x^2+y^2-16x-18y-4=0$
3. $x^2+y^2+16x+18y+12=0$
4. $x^2+y^2+16x-18y+4=0$

Q. If the two circles $2x^2+2y^2-3x+6y+k=0$ and $x^2+y^2-4x+10y+16=0$ cut orthogonally, then the value of $k$ is .

1. $4$
2. $10$
3. $1$
4. $5$

Q. If the two circles, which pass through $(0,a)$ and $(0,-a)$ and touch the line $y=mx+c$, will cut orthogonally, if

1. $c^2=a^2(2-m^2)$
2. $\frac{a^2}{c^2}=\frac{1}{2+m^2}$
3. $c^2=a^2(2+m^2)$.
4. $c^2=a^2(1-m^2)$.

Q. Consider the circles $S_1:x^2+y^2=4$ and $S_2:x^2+y^2-2x-4y+4=0.$ Which of the following statements is (are) CORRECT?

1. Number of common tangents to these circles is $2.$
2. If the power of a variable point $P$ with respect to these two circles is same, then $P$ moves on the line $x+2y-4=0$.
3. Sum of the $y-$intercepts of both the circles is $6$.
4. The circles $S_1$ and $S_2$ are orthogonal.

Q. The equation of circle touching the line $2x+3y+1=0$ at $(1,-1)$ and cutting orthogonally the cirlce having line segment joining $(0,3)$ and $(-2,-1)$ as diameter is

1. $2x^2+2y^2+10x+5y+1=0$
2. $2x^2+2y^2-10x-5y+1=0$
3. $2x^2+2y^2-10x-10y+10=0$
4. $2x^2+2y^2-10x-10y+20=0$

Q. Let $f$ be a differentiable function such that $f^{\prime }\left(x\right)=7-\frac{3}{4}\cdot \frac{f\left(x\right)}{x},(x>0)$ and $f\left(1\right)\ne 4$. Then $\underset{x\to 0^{+}}{lim}x\cdot f\left(\frac{1}{x}\right)$ :

1. exists and equals $4$.
2. does not exist.
3. exists and equals $\frac{4}{7}.$
4. exists and equals $0.$

Q. The members of a family of circles are given by the equation $2(x^2+y^2)+\lambda x-(1+{\lambda }^2)y-10=0$. The number of circle(s) from this family that is/are cut orthogonally by the circle $x^2+y^2+4x+6y+3=0$ is

1. $2$
2. infinite
3. $1$
4. $0$

Q. Two congruent circles with centres at $(2,3)$ and $(5,6)$, which intersect at right angles, have radius equal to

1. None of these
2. $2\sqrt{2}$
3. $3$
4. $4$

Q.

If $x^2+y^2+px+3y-5=0$ and $x^2+y^2+5x+py+7=0$ cut orthogonally, then p is

1. 2

2. 

3. 1

4. 

Q. Consider two circles passing through two distinct points $(0,a)$ and $(0,-a)$ on $y$-axis and touching the line $y=3x+4$. If the circles are orthogonal and $a=\pm \frac{k_1}{\sqrt{k_2}}$, where $k_1,k_2$ are co-prime, then the value of $k_1+k_2$ is

Q. A circle $S$ passes through the point $(0,1)$ and is orthogonal to the circles $(x-1{)}^2+y^2=16$ and $x^2+y^2=1$. Then

1. radius of $S$ is $8$
2. radius of $S$ is $7$
3. centre of $S$ is $(-7,1)$
4. centre of $S$ is $(-8,1)$

Q. The equation of the circle which orthogonally intersects the circles $x^2+y^2-2x+3y-7=0$, $x^2+y^2+5x-5y+9=0$ and $x^2+y^2+7x-9y+29=0$, is

1. $x^2+y^2-16x-8y-12=0$
2. $x^2+y^2-16x-18y-4=0$
3. $x^2+y^2+16x-18y+4=0$
4. $x^2+y^2+16x+18y+12=0$

Q. The equation of circle touching the line $2x+3y+1=0$ at $(1,-1)$ and orthogonally cutting the cirlce whose endpoints of diameter are $(0,3)$ and $(-2,-1)$ is

1. $2x^2+2y^2+10x+5y+1=0$
2. $2x^2+2y^2-10x-5y+1=0$
3. $2x^2+2y^2-10x-10y+10=0$
4. $2x^2+2y^2-10x-10y+20=0$

Q. Let $S$ be the circle which cuts the three circles $x^2+y^2-3x-6y+14=0,x^2+y^2-x-4y+8=0$ and $x^2+y^2+2x-6y+9=0$ orthogonally. Then

1. From the point $(2,3)$, two tangents can be drawn to $S$
2. The radius of $S$ is $2$
3. The absolute difference between the coordinates of the center of $S$ is $1$
4. The center of $S$ is $(1,2)$

Q. Consider the circles $S_1:x^2+y^2=4$ and $S_2:x^2+y^2-2x-4y+4=0.$ Which of the following statements is (are) CORRECT?

1. Number of common tangents to these circles is $2.$
2. If the power of a variable point $P$ with respect to these two circles is same, then $P$ moves on the line $x+2y-4=0$.
3. Sum of the $y-$intercepts of both the circles is $6$.
4. The circles $S_1$ and $S_2$ are orthogonal.

Q. Two congruent circles with centres at $(2,3)$ and $(5,6),$ which intersect at right angles, have radius equal to

1. $2\sqrt{2}$
2. $3$
3. $4$
4. None of these

Q. A circle $S$ passes through the point $(0,1)$ and is orthogonal to the circles $(x-1{)}^2+y^2=16$ and $x^2+y^2=1$, then

1. centre of $S$ is $(-7,1)$
2. centre of $S$ is $(-8,1)$
3. radius of $S$ is $7$
4. radius of $S$ is $8$

Q. Find the area bouded by the curves $y=2x-x^2,4y=(x-2{)}^2$ and $y=0$

1. $\frac{12}{25}sq.units.$
2. $\frac{1}{25}sq.units.$
3. $\frac{12}{375}sq.units.$
4. $\frac{36}{75}sq.units.$

Q. The equation of circle touching the line $2x+3y+1=0$ at $(1,-1)$ and cutting orthogonally the circle having line segment joining $(0,3)$ and $(-2,-1)$ as diameter is

1. $2x^2+2y^2+10x+5y+1=0$
2. $2x^2+2y^2-10x-5y+1=0$
3. $2x^2+2y^2-10x-10y+10=0$
4. $2x^2+2y^2-10x-10y+20=0$

Q. If a circle passes through the point (a, b) and cuts the circle $x^2+y^2=4$ orthogonally, then the locus of its centre is

1. $2ax+2by+(a^2+b^2+4=0)$
2. $2ax+2by-(a^2+b^2+4)=0$
3. $2ax-2by-(a^2+b^2+4)=0$
4. $2ax-2by-(a^2+b^2+4)=0$

Q. The locus of the centre of the circle which cuts $x^2+y^2-20x+4=0$ orthogonally and touches the line $x=2,$ is

1. $y^2=4(4x+1)$
2. $y^2=16x$
3. $x^2=4(4y+1)$
4. $y^2=-16x$

Q. A plane meet the co-ordinates axes in $A,B,C$ such that the centroid of triangle $ABC$ is the point $\alpha ,\beta ,\gamma .$ If the equation of the plane be $\frac{x}{\alpha }+\frac{y}{\beta }+\frac{z}{\gamma }=k$ then, $k=?$

1. $1$
2. $3$
3. $2$
4. ${\alpha }^2+{\beta }^2+{\gamma }^2$

Q. The equation of the circle which cuts orthogonally each of the three circles given below:
$x^2+y^2-2x+3y-7=0$, $x^2+y^2+5x-5y+9=0$ and $x^2+y^2+7x-9y+29=0$

1. $x^2+y^2-16x-8y-12=0$
2. $x^2+y^2-16x-18y-4=0$
3. $x^2+y^2+16x-18y+4=0$
4. $x^2+y^2+16x+18y+12=0$

Q. Find the equation of locus of a point in which the line segment $A(1,2)$ and $B(-3,4)$ is ${90}^o$

Q. Consider two circles passing through two distinct points $(0,a)$ and $(0,-a)$ on $y$-axis and touching the line $y=3x+4$. If the circles are orthogonal and $a=\pm \frac{k_1}{\sqrt{k_2}}$, where $k_1,k_2$ are co-prime, then the value of $k_1+k_2$ is

Q. The equation of the circle which orthogonally intersects the circles $x^2+y^2-2x+3y-7=0$, $x^2+y^2+5x-5y+9=0$ and $x^2+y^2+7x-9y+29=0$, is

1. $x^2+y^2-16x-8y-12=0$
2. $x^2+y^2-16x-18y-4=0$
3. $x^2+y^2+16x-18y+4=0$
4. $x^2+y^2+16x+18y+12=0$

Q. The equation of the circle which orthogonally intersects the circles $x^2+y^2-2x+3y-7=0$, $x^2+y^2+5x-5y+9=0$ and $x^2+y^2+7x-9y+29=0$, is

1. $x^2+y^2-16x-8y-12=0$
2. $x^2+y^2-16x-18y-4=0$
3. $x^2+y^2+16x-18y+4=0$
4. $x^2+y^2+16x+18y+12=0$

Q. Let $P$ be the point on the parabola, $y^2=8x$ which is at a minimum distance from the centre $C$ of the circle, $x^2+{(y+6)}^2=1$. Then the equation of the circle, passing through $C$ and having its centre at $P$ is:

1. $x^2+y^2-4x+8y+12=0$
2. $x^2+y^2-x+4y-12=0$
3. $x^2+y^2-\frac{x}{4}+2y-24=0$
4. $x^2+y^2-4x+9y+18=0$

Q. The intercept on the line $y=x$ by the circle $x^2+y^2-2x=0$ is AB. Equation of the circle on AB as a diameter is:

1. $x^2+y^2-x-y=0$
2. $x^2+y^2-x+y=0$
3. $x^2+y^2+x+y=0$
4. $x^2+y^2+x-y=0$

Q. If the two circles, which pass through $(0,a)$ and $(0,-a)$ and touch the line $y=mx+c$, will cut orthogonally, if

1. $c^2=a^2(2-m^2)$
2. $\frac{a^2}{c^2}=\frac{1}{2+m^2}$
3. $c^2=a^2(2+m^2)$.
4. $c^2=a^2(1-m^2)$.