Circles practice · Mixed difficulty

The locus of the centre of a circle of radius 2 which rolls on the outside of the circle, is ${x^2} + {y^2} + 3x - 6y - 9 = 0$ is

The range of values of $\theta \in \left[ {0,2\pi } \right]$ for which $\left( {1 + \cos \theta ,\sin \theta } \right)$ is on interior point of the circle ${x^2} + {y^2} = 1,$ is

The equation of circle which touches the line $x+3 y-2=0$ and $3 x+9 y-2=0$, also the point of contact on first line is $(-1,1)$, is

Length of the common chord of the circles $(x-1)^2+(y+1)^2=c^2$ and $(x+1)^2+(y-1)^2=c^2$ is

The tangent to ${x^2} + {y^2} = 9$ which is parallel to $y - axis$ and does not lie in the third quadrant touches the circle at the point

The locus of a point which moves such that the sum of the squares of its distances from three vertices of a triangle is a circle whose centre is at the

If the points $(0,0),(1,0),(0,1)$ and $(t, t)$ are concyclic, then $t=$

Find the equation of the normal to the circle $x^2+y^2-5 x+2 y$ $-48=0$ at the point $(5,6)$.

The locus of the centre of a circle which touches externally the circle $x^2+y^2-6 x-6 y+14=0$ and also touches $Y$-axis, is given by the equation

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

The equation of a circle with origin as centre and passing through the vertices of an equilateral triangle whose median is of length 3a is

The distance between the chords of contact of the tangent to the circle ${x^2} + {y^2} + 2gx + 2fy + c = 0$ from the origin and the point $\left( {g,f} \right),$ is

The set of values of a for which the point $(2 a, a+1)$ is an interior point of the larger segment of the circle $x^2+y^2-2 x-2 y-8=0$ made by the chord $x-y+1=0$, is

The pole of the line $3 x+5 y+17=0$ with respect to the circle $x^2+y^2+4 x+6 y+9=0$ is

The normal at the point $\left( {3,4} \right)$ on a circle at the point $\left( { - 1, - 2} \right).$ The equation of the circle, is

If the line $y-m x+m-1=0$ cuts the circle $x^2+y^2-4 x-4 y+4$ $=0$ at two real points, then $m$ belongs to

If the circles ${\left( {x - a} \right)^2} + {\left( {y - b} \right)^2} = {c^2}$ and ${\left( {x - b} \right)^2} + {\left( {y - a} \right)^2} = {c^2}$ touch each other, then

The circles ${x^2} + {y^2} - 8x + 4y + 4 = 0$ touches

The equation $x^2+y^2-6 x+8 y+25=0$ represents

The equation of the locus of the middle point of a chord of the circle $x^2+y^2-2 x-2 y=0$ such that the pair of lines joining the origin to the point of intersection of the chord and the circle are equally inclined to the $x$-axis is

The number of common tangents of the circles given by $x^{2}+y^{2}-8 x-2 y+1=0$ and $x^{2}+y^{2}+6 x+8 y=0$ is

If three distinct point $A, B, C$ are given in the 2-dimensional coordinate plane such that the ratio of the distance of each one of them from the point $(1,0)$ to the distance from $(-1,0)$ is equal to $\frac{1}{2}$, then the circumcentre of the triangle $A B C$ is at the point:

The condition for a line $y = 2x + c$ to touch the circle ${x^2} + {y^2} = 16$ is

The circle ${x^2} + {y^2} - 3x - 4y + 2 = 0$ cuts $x$-axis at

The lines $5 x+12 y-10=0$ and $5 x-12 y-40=0$ touch a circle of radius 3 units. If the centre of circle lies in the first quadrant, then the coordinates of centre is

The point of the straight line $y = 2x + 11$ which is nearest to the circle $16\left( {{x^2} + {y^2}} \right) + 32x - 8y - 50 = 0,$ is

If $5x - 12y + 10 = 0$ and $12y - 5x + 16 = 0$ are two tangents to a circle, the radius of the circle is

If the area of the circle $4{x^2} + 4{y^2} - 8x + 16y + k = 0$ is $9\pi$ sq unit, then the value of $k$ is

If the line $x \cos \theta+y \sin \theta=2$ is the equation of a transverse common tangent to the circles $x^2+y^2=4$ and $x^2+y^2-6 \sqrt{3} x-6 y+20=0$, then the value of $\theta$ is

If two circles $x^2+y^2+2 g x+2 f y=0$ and $x^2+y^2+2 g_1 x+2 f_1 y=0$ touch each other, then