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

Q. A square is inscribed in the circle $x^2+y^2-2x+4y+3=0,$ whose sides are parallel to the coordinate axes. One vertex of the square is

1. $(1+\sqrt{2},-2)$
2. $(1-\sqrt{2},-2)$
3. $(1-2+\sqrt{2})$
4. None of these

Q. Centre of the circle $(x-3{)}^2+(y-4{)}^2$ = 5 is

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

Q.

An infinite number of tangents can be drawn from (1, 2) to the circle $x^2+y^2-2x-4y+\lambda =0,$ then $\lambda =$

1. -20

2. 0

3. 5

4. Cannot be determined

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 g

2. A circle of radius f

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

4. A circle of radius 0

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. The radius of circle $x^2+y^2-12x+8y+32=0,$ is

1. $2\sqrt{5}$
2. $\sqrt{10}$
3. $4\sqrt{11}$
4. $\sqrt{5}$

Q. The circle passing through the intersection of the circles, $x^2+y^2-6x=0$ and $x^2+y^2-4y=0$, having its centre on the line, $2x-3y+12=0$, also passes through the point

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.

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. If $k{x}^2+k{y}^2-4x-6y-2k=0$ represents a real circle, then the set of values of $k$ is

1. $(0,\infty )$
2. $(-\infty ,0)$
3. $R$
4. $R-\left\{0\right\}$

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.

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

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

Q. Let $C$ be a circle passing through the origin and making an intercept of $\sqrt{10}$ on the line $y=2x+\frac{5}{\sqrt{2}}$. If the line subtends an angle of ${45}^{\circ }$ at the origin, then the equation of circle $C$ is/are

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

Q. If the line x + 2by + 7 = 0 is diameter of the circle $x^2+y^2-6x+2y=0$, then b =

1. 3
2. -5
3. -1
4. 5

Q. If $2x^2+2y^2-12x+8y+k=0$ is a point circle, then the value of $k$ is

1. $18$
2. $12$
3. $13$
4. $26$

Q. The equation of the image of circle $x^2+y^2-4x+6y+10=0$ in the line $4x-3y+8=0$ is

1. $(x-6{)}^2+(y-3{)}^2=3$
2. $(x+6{)}^2+(y-3{)}^2=3$
3. $(x+6{)}^2+(y+3{)}^2=3$
4. $(x-6{)}^2+(y+3{)}^2=3$