Obtaining centre and radius of a circle from general equation of a circle

Q. If $g^2+f^2=c,$ then the equation $x^2+y^2+2gx+2fy+c=0$ will represent

1. A circle of radius f

2. A circle of radius g

3. A circle of radius 0

4. A circle of diameter $\sqrt{c}$

Q.

The centre of the circle, whose radius is $5$ and which touches the circle $x^2+y^2-2x-4y-20=0$ at $\left(5,5\right)$ is

1. $\left(10,5\right)$

2. $\left(5,8\right)$

3. $\left(5,10\right)$

4. $\left(8,9\right)$

5. $\left(9,8\right)$

Q. The radius of the circle passing through the point (6, 2) and two of whose diameters are x + y = 6 and x + 2y = 4 is

1. 4

2. 6

3. 20
4. $\sqrt{20}$

Q. If the two circles $(x-1{)}^2+(y-3{)}^2=r^2$ and $x^2+y^2-8x+2y+8=0$ intersect at two distinct points, then

1. r < 2
2. r = 2
3. 2 < r < 8
4. r > 2

Q.

The number of integral values of $\lambda$ for which $x^2+y^2+\lambda x+(1-\lambda )y+5=0$ is the equation of a circle whose radius cannot exceed 5, is

1. 14

2. 18

3. 16

4. None of these

Q. If OA, OB are two equal chords of the circle $x^2+y^2-2x+4y=0$ perpendicular to each other and passing through the origin, then the equations of OA and OB are

1. 3x + y = 0, x + 3y = 0
2. 3x – y = 0, x – 3y = 0
3. 3x – y = 0, x + 3y = 0
4. 3x + y = 0, x – 3y = 0

Q. The area of the circle whose centre is at (1, 2) and which passes through the point (4, 6) is

1. $5\pi$
2. $10\pi$
3. $25\pi$
4. None of these

Q. A circle $x^2+y^2+2gx+2fy+c=0$ passing through (4, 2) is concentric to the circle $x^2+y^2-2x+4y+20=0,$ then the value of c will be

1. -4
2. 4
3. 0
4. 1

Q.

If a chord of the circle $x^2+y^2-4x-2y-c=0$ is trisected at the points (1/3, 1/3) and (8/3, 8/3), then

1. Length of the chord=$7\sqrt{2}$
   

2. c=20
   

3. Radius of the circle 25
   

4. c=25

Q. If two distinct chords drawn from the point (p, q) on the circle $x^2+y^2-px-qy=0(wherepq\ne 0)$ are bisected by the x-axis, then

1. $p^2=q^2$
2. $p^2=8q^2$
3. $p^2<8q^2$
4. $p^2>8q^2$

Q. Radius of the circle $x^2+y^2+2xcos\theta +2ysin\theta -8=0,$ is

1. 1
2. 3
3. $2\sqrt{3}$
4. $\sqrt{10}$

Q. The equation $x^2+y^2=0$ denotes

1. A circle

2. A point

3. y-axis
4. x-axis

Q. The centre and radius of the circle $2x^2+2y^2-x=0$ are

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

Q. The radius of the circle $x^2+y^2+4x+6y+13=0$ is

1. 0
2. $\sqrt{13}$
3. $\sqrt{26}$
4. $\sqrt{23}$

Q. The equation of a diameter of circle $x^2+y^2+6x+2y=0$ passing through origin is

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

Q. Centre of circle passing through $A(0,1),B(2,3),C(-2,5)$ is

1. $(-1,10)$
2. $(\frac{1}{3},\frac{10}{3})$
3. $(\frac{-1}{3},\frac{10}{3})$
4. $(\frac{10}{3},\frac{-2}{3})$

Q.

Find the radius of the circle $x^2+y^2-2x+4y-11=0$

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Q. The area of the curve $x^2+y^2=2ax$ is

1. $4\pi a^2$
2. $\pi a^2$
3. $2\pi a^2$
4. $\frac{1}{2}\pi a^2$

Q. If the radius of the circle $x^2+y^2-18x+12y+k=0$ be 11, then k =

1. 347
2. 4
3. -4
4. 49