Properties of Orthogonality of Two Circles

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. Center of S is (-8, 1)

Q. A circle $C_1$ passes through the origin and has its centre on the line $y=x$. Let $C_1$ cuts $C_2:x^2+y^2-4x-6y+10=0$ orthogonally. If the radius of $C_1$ is $r$, then the value of $8r^2$ is

Q.

Find the equation of the circle orthogonal to the circles $x^2+y^2+3x-5y+6=0and4x^2+4y2-28x+29=0$ and whose center lies on the line 3x + 4y + 1 = 0.

1. x² + y² + y/4 - 29/4 = 0

2. 2x² + 2y² + y - 29 = 0

3. 8x² + 8y² + 2y - 29 = 0

4. 4x² + 4y² + 2y - 29 = 0

Q.

If a circle passes through the point (1, 2) and cuts the circle $x^2+y^2=4$ orthogonally, then the equation of the locus of its centre is

1. 2x + 4y - 1 = 0

2. 

3. 

4. 2x + 4y - 9 = 0

Q. The centre of the circles $x^2-y^2-6x-8y-7=0$ and $x^2-y^2-4x-10y-3=0$ are the ends of the diameter of the circle:

1. $x^2-y^2-5x-9y-26=0$
2. $x^2-y^2-5x-9y+26=0$
3. $x^2+y^2+5x-y-14=0$
4. $x^2+y^2+5x+y+14=0$

Q. If the circles $x^2+y^2-8x-6y+21=0$ and $x^2+y^2+ay-15=0$ are orthogonal then the value of $a$ is

1. $-2$
2. $-4$
3. $-\frac{1}{2}$
4. $-1$

Q. Find the area of the smaller region bounded by the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ and the line $\frac{x}{a}+\frac{y}{b}=1$

Q. Insert $2$ geometric means between $3$ and $81$.

Q. Two given circles $x^2+y^2+ax+by+c=0$ and $x^2+y^2+dx+ey+f=0$ will intersect each other orthogonally, only when

1. ad+be = c+f
2. 2ad+2be = c+f
3. a+b+c = d+e+f

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. Center of S is (-8, 1)

Q. Let $S_n$ be the director circle of the circle $S_{n-1}$, where $n\ge 2$. $S_1:x^2+y^2=1$. The equation of the circle orthogonal to the circle $S_{10}$ is given by $x^2+y^2+2gx+2fy+c=0$. Then

1. the value of $c$ cannot be determined
2. no such circle is possible
3. there is a unique value of $c$
4. the value of $c$ depends upon $g$ and $f$

Q. The point $P(3,3)$ is reflected across the line $y=-x$. Then it is translated horizontally $3$ units to the left and vertically $3$ units up. Finally it is reflected across the line $y=x$. What are the co-ordinates of the point after these transformations.

1. $(0,-6)$
2. $(-6,6)$
3. $(0,0)$
4. $(-6,0)$

Q. The area bounded by $y^2=4x$ and $x^2=4y$ is

1. $\frac{20}{3}$ sq. units
2. $\frac{16}{3}$ sq. units
3. $\frac{10}{3}$ sq. units
4. $\frac{14}{3}$ sq. units

Q. A point on the parabola $y^2=18x$ at which the ordinate increase at twice the rate of the abscissa is

1. $(9/8,9/2)$
2. $(2,-4)$
3. $(-9/8,9/2)$
4. $(2,4)$

Q. $A$ is a point on $y^2=4ax$. The normal at $A$ meets the parabola again at $B$. If $AB$ subtends a right angle at its vertex then the slope of $AB$ is

1. $\frac{1}{2}$
2. $\frac{-1}{2}$
3. $\frac{2}{3}$
4. $\sqrt{2}$

Q.

Two circles $x^2+y^2+2g_{1}x+2f_{1}y+c_1=0and{x}^2+y^2+2g_{2}x+2f_{2}y+c^2=0$ are said to be orthogonal. Then $2g_1{g}_2+2f_1{f}_2=c_1+c_2$

1. True

2. False

Q. A circle has the equation $x^2+y^2=16$. How do you find the center, radius and the intercepts?

Q. $f\left(x\right)={\cot }^{-1}\left(\frac{x}{\sqrt{x^2-\left[x^2\right]}}\right),x\in R$ is

1. $R-\{\sqrt{n},n\ge 0,n\in I\}$
2. $R$
3. $R-0$
4. $R-\{\pm \sqrt{n},n\in N\}$

Q. The centre of the circle $\left(s\right)$ passing through the points $(0,0),(1,0)$ and touching the circle $x^2+y^2=9,$ is (are)

1. $(3/2,1/2)$
2. $(1/2,3/2)$
3. $(1/2,-2^{1/2})$
4. $(1/2,2^{1/2})$

Q. I. If the points $(a,0)$ , $(b,0)$ , $(0,c)$ , $(0,d)$ are concyclic, then $ab=cd$
II. If the points $(1,-6),(5,2),(7,0),(-1,k)$ are concyclic then $k=-3$.

1. Only I is true
2. Only II is true
3. I & II are true
4. Neither I & II are true

Q. Find the radius of the circle which passes through the origin, $(0,4)$ and $(4,0)$.

1. $\sqrt{8}$
2. $4$
3. $16$
4. $\sqrt{36}$

Q. If the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(b>a)$ and the parabola $y^2=4ax$ cut at right angles, then eccentricity of the ellipse is:

1. $\frac{3}{5}$
2. $\frac{2}{3}$
3. $\frac{1}{\sqrt{2}}$
4. $\frac{1}{2}$

Q.

Two circles $x^2+y^2+2g_{1}x+2f_{1}y+c_1=0and{x}^2+y^2+2g_{2}x+2f_{2}y+c^2=0$ are said to be orthogonal. Then $2g_1{g}_2+2f_1{f}_2=c_1+c_2$

1. True

2. False

Q. If the eccentricity of the ellipse $\frac{x^2}{{\left(\log a\right)}^2}+\frac{y^2}{{\left(\log b\right)}^2}=1(a>b>0,a\ne 1)$ is $\frac{1}{\sqrt{2}}$ and $c$ be the eccentricity of the hyperbola $\frac{x^2}{{\left({\log }_{b}a\right)}^2}-y^2=1$ then $e^2$ is greater than (where $\log x-\ln x$)

1. $\frac{3}{2}$
2. $\frac{1}{2}$
3. $\frac{2}{3}$
4. $\frac{5}{4}$

Q. Find the equation of the circle having radius 5 and which touches line 3x +4y - 11= 0 at point (1, 2).

Q. Two given circles $x^2+y^2+ax+by+c=0$ and $x^2+y^2+dx+ey+f=0$ will intersect each other orthogonally, only when

1. a+b+c = d+e+f
2. ad+be = c+f
3. 2ad+2be = c+f

Q. Two given circles $x^2+y^2+ax+by+c=0$ and $x^2+y^2+dx+ey+f=0$ will intersect each other orthogonally, only when

1. a+b+c = d+e+f
2. ad+be = c+f
3. 2ad+2be = c+f

Q.

Find the equation of the circle orthogonal to the circles $x^2+y^2+3x-5y+6=0and4x^2+4y2-28x+29=0$ and whose center lies on the line 3x + 4y + 1 = 0.

1. $x^2+y^2+\frac{y}{4}-\frac{29}{4}=0$

2. $2x^2+2y^2+y-29=0$

3. $8x^2+8y^2+2y-29=0$

4. $4x^2+4y^2+2y-29=0$

Q. Find the equation of the circle which cuts the circles
$x^2+y^2-9x+14=0$
and $x_2+y^2+15x+14=0$
orthogonally and passes through the point $(2,5)$.