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

Q. The equation $x^2+y^2+4x+6y+13=0$ represents

1. Pair of coincident straight lines

2. Circle

3. Pair of concurrent straight lines

4. Point

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

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

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. The centres of the circles $x^2+y^2=1,x^2+y^2+6x-2y=1$ and $x^2+y^2-12x+4y=1$ are

1. Same
2. Collinear
3. Non-collinear
4. None of these

Q. The equation of the circle which passes through the points $(1,0),(0,-6)$ and $(3,4)$ is

1. $4x^2+4y^2-142x-47y+138=0$
2. $4x^2+4y^2+142x+47y+140=0$
3. $4x^2+4y^2-142x+47y+138=0$
4. $4x^2+4y^2+150x-49y+138=0$

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

Q. The circle passing through the distinct points $\left(1,t\right),\left(t,1\right)$ and $\left(t,t\right)$ for all values of ${}^{\prime }{t}^{\prime }$ , passes through the point:

1. $\left(-1,1\right)$
2. $\left(1,1\right)$
3. $\left(-1,-1\right)$
4. $\left(1,-1\right)$

Q. The equation of the circle circumscribing the triangle formed by the lines $x+y=6$, $2x+y=4$ and $x+2y=5$ is:

1. $x^2+y^2+17x+19y-50=0$
2. $x^2+y^2-17x-19y-50=0$
3. $x^2+y^2+17x-19y-50=0$
4. $x^2+y^2-17x-19y+50=0$

Q. The length of the diameter of the circle $x^2+y^2-4x-6y+4=0$ is -

1. $3$
2. $9$
3. $4$
4. $6$

Q. The value of $k$ for which two tangents can be drawn from $(k,k)$ to the circle $x^2+y^2+2x+2y16=0$ is

1. $kϵR^{+}$
2. $kϵR$
3. $kϵ(-\infty ,-4)\cup (2,\infty )$
4. $kϵ(0,1]$