Obtaining Centre and Radius of a Circle from General Equation of a Circle

Q.

A square is inscribed in the circle $x^2+y^2-6x+8y-103=0$ with its sides parallel to the coordinate axes. Then the distance of the vertex of this square which is nearest to the origin is:

1. $6$
2. $\sqrt{41}$
3. $13$
4. $\sqrt{137}$

Q. If the area of an equilateral triangle inscribed in the circle, $x^2+y^2+10x+12y+c=0$ is $27\sqrt{3}$ sq. units then $c$ is equal to :

1. $13$
2. $20$
3. $25$
4. $-25$

Q. The equation of the circle passsing through $(1,0)$ and $(0,1)$ and having the smallest possible radius is

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

Q. The equation of the circle passing through the points (0, 0), (0, b) and (a, b) is

1. 
2. 
3. 
4. 

Q.

If $2x^2+\lambda xy+2y^2+(\lambda -4)x+6y-5=0$
is the equation of a circle, then its radius is

1. $3\sqrt{2}$

2. $2\sqrt{2}$

3. None of these

4. $2\sqrt{3}$

Q.

If the equation $(4a-3)x^2+a{y}^2+6x-2y+2=0$ represents a circle, then its centre is

1. (3, -1)

2. (3, 1)

3. (-3, -1)

4. None of these

Q. If the curves $x^2-6x+y^2+8=0$ and $x^2-8y+y^2+16-k=0,(k>0)$ touch each other at a point, then the largest value of $k$ is

Q.

The area (in sq. units) of the part of the circle ${\mathrm{x}}^2+{\mathrm{y}}^2=36$, which is outside the parabola ${\mathrm{y}}^2=9\mathrm{x}$, is:

1. $24\mathrm{\pi }+3\sqrt{3}$

2. $12\mathrm{\pi }+3\sqrt{3}$

3. $12\mathrm{\pi }-3\sqrt{3}$

4. $24\mathrm{\pi }-3\sqrt{3}$

Q. The equation of the circle with centre on the x-axis, radius 4 and passing through the origin, is

1. 
2. 
3. 
4. 

Q. The circle represented by the equation $x^2+y^2+2gx+2fy+c=0$ will be a point circle, if

1. 
2. 
3. None of these
4. x-axis at the origin

Q. The equation of the circle passing through three points $(1,0),$ $(-1,0)$ and $(0,1),$ is

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

Q. If the circle $x^2+y^2-6x-10y+c=0$ does not touch or intersect the coordinate axes and $(1,4)$ lies inside the circle, then the number of integral values of $c$ is

Q.

If the circles $x^2+y^2=a$ and $x^2+y^2-6x-8y+9=0$, touch externally, then a=

1. 1

2. -1

3. 21

4. 16

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 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

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

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

Q. The equation of the circle passing through the origin and cutting intercepts of length 3 and 4 units from the positive axes, is

1. 
2. 
3. 
4. 

Q. The equation of the circle having centre (1, –2) and passing through the point of intersection of lines 3x + y = 14, 2x + 5y = 18 is

1. 
2. 
3. 
4. 

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

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

Q.

The coordinates of the centre of a circle, whose radius is $2$ units and which touches the line pair $x^2-y^2-2x+1=0,$ are

1. $(4,0)$
2. $(1+2\sqrt{2},0)$
3. $(4,1)$
4. $(1,2\sqrt{2})$

Q.

The area of an equilateral triangle inscribed in the circle $x^2+y^2-6x-8y-25=0$ is

1. $25\pi$

2. None of these

3. $\frac{225\sqrt{3}}{6}$

4. $50\pi -100$

Q.

$(1,4)$ and $(3,8)$ are the end points of diameter of a circle. Find the radius and center of this circle.

1. 2√5 and (2, 6)

2. √5 and (2, 6)

3. √5 and (6, 2)

4. 2√5 and (6, 2)

Q.

If the circles $x^2+y^2=9$ and $x^2+y^2++8y+c=0$
touch each other, then c is equal to

1. -15

2. -16

3. 16

4. 15

Q.

Find the coordinates of the centre and radius of each of the following circles :

(i) $x^2+y^2+6x-8y-24=0$
(ii) $2x^2+2y^2-3x+5y=7$
(iii) $\frac{1}{2}(x^2+y^2)+xcos\theta +ysin\theta -4=0.$
(iv) $x^2+y^2-ax-by=0$

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.

Prove that the radii of the circles
$x^2+y^2=1,x^2+y^2-2x-6y-6=0$
and $x^2+y^2-4x-12y-9=0$ are in A.P

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. \N
4. 1

Q.

The equation $x^2+y^2+2x-4y\times 5=0$ represents

1. none of these

2. a point

3. a circle of non-zero radius

4. a pair of straight lines

Q.

The radius of the circle representd by the equation $3x^2+3y^2+\lambda xy+9x+(\lambda -6)y+3=0$

1. None of these

2. $\frac{3}{2}$

3. $\frac{\sqrt{17}}{2}$

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

Q.

If (-3, 2) lies on the circle $x^2+y^2+2gx+2fy+c=0$ which is concentric with the circle $x^2+y^2+6x+8y-5=0$, then c=

1. 11

2. 24

3. None of these

4. -11