Circles practice · Mixed difficulty

The equation of the tangent from the point $\left( {0,1} \right)$ to the circle ${x^2} + {y^2} - 2x - 6y + 6 = 0$, is

The angular between the tangent drawn from the origin to the circle ${\left( {x - 7} \right)^2} + {\left( {y + 1} \right)^2} = 25$ is

Circles are drawn through the point $(3,0)$ to cut an intercept of length 6 units on the negative direction of the $x$-axis. The equation of the locus of their centres, is

If the straight line $x - 2y + 1 = 0$ intersects the circle ${x^2} + {y^2} = 25$ in points $P$ and $Q$, then the coordinates of the point of intersection of tangents drawn at $P$ and $Q$ to the circle ${x^2} + {y^2} = 25$ are

If a square with length of each side equal to $a$ is inscribed in the circle $x^2+y^2+4 x+10 y+21=0$, then $a$ is equal to

The number of common tangents to the circles ${x^2} + {y^2} - x = 0,{x^2} + {y^2} + x = 0$ is

The circles $x^2+y^2=4$ and $x^2+y^2-4 \lambda x+9=0$ have two common tangents if

If the lengths of the tangents from two points $A$ and $B$ to a circle are $l$ and $l^{\prime}$ respectively and points are conjugate with respect to the circle (i.e. each passes through the polar of the other with respect to the circle), then $(A B)^2=$

The value of $k$ for which two tangents can be drawn from $(k, k)$ to the circle $x^2+y^2+2 x+2 y-16=0$ is

The set of values of '$c$' so that the equation $y = \left| x \right| + c$ and ${x^2} + {y^2} - 8\left| x \right| - 9 = 0$ have no solution, is

The locus of the point of intersection of perpendicular tangents to the circles $x^2+y^2=a^2$ and $x^2+y^2=b^2$, is

The locus of a point such that the tangents drawn from it to the circle $x^2+y^2-6 x-8 y=0$ are perpendicular to each other is

If the chord of contact of tangents from a point $P\left( {{x_1},{y_1}} \right)$ to the circle ${x^2} + {y^2} = {a^2}$ touches the circle ${\left( {x - a} \right)^2} + {y^2} = {a^2},$ then the locus of $\left( {{x_1},{y_1}} \right),$ is

The equation of normal of ${x^2} + {y^2} - 2x + 4y - 5 = 0$ at $\left( {2,1} \right)$ is

Let $S \equiv 0$ and $S^{\prime} \equiv 0$ be two circles belonging to the family of circles $x^2+y^2+\lambda x-4 y+3=0:(\lambda$ $\in \mathrm{R})$ such that $S \equiv 0$ bisects the circumference of circle $S^{\prime} \equiv 0$ and has radius equal to twice of radius of ${S}^{\prime} \equiv 0$. Then distance between their centres is

The circle $x^2+y^2-2 x-3 k y-2=0$ passes through some fixed point. One of them may be:

The equation of sphere concentric with the sphere $x^2+y^2+z^2-4 x-6 y-8 z-5=0$ and which passes through the point $(0,1,0)$ is

One hundred concentric circles with radii $1,2,3-100$ are drawn in a plane. The interior of circle with radius 1 is colored red & each region bounded by consecutive circles is coloured either red or green with no two adjacent regions the same color, then sum of total area of green regions will be

The tangents drawn from the origin to the circle $x^2+y^2-2 r x-2 h y+h^2=0$ are perpendicular, if

If the lines $2x + 3y + 1 = 0$ and $3x - y - 4 = 0$ lie along diameters of a circle of circumference $10\pi$, then the equation of the circle is

Line $(x-3) \cos \theta+(y-3) \sin \theta=1$ touches a circle $(x-3)^2+(y-3)^2=1$, then find the number of values of $\theta$.

The centre and radius of the circle $3 x^2+3 y^2-8 x-10 y+3=0$ are

The locus of the point $(\sqrt{3 h+2}, \sqrt{3 k})$. If $(h, k)$ lies on $x+y=1$ is

A square is inscribed in the circle $x^2+y^2-2 x+4 y-93=0$ with its sides parallel to the axes of coordinates. The coordinates of the vertex farthest from the origin are

A variable circle $C$ has the equation $x^2+y^2-2\left(t^2-3 t+1\right) x-2\left(t^2+2 t\right) y+t=0$, where $t$ is $a$ parameter. If the power of point $(a, b)$ w.r.t. the circle $C$ is constant, then the ordered pair $(a, b)$ is

The abscissae of two points $A$ and $B$ are the roots of the equation ${x^2} - 2ax - {b^2} = 0,$ and their ordinates are the roots of the equation ${x^2} + 2px - {q^2} = 0.$ The radius of the circle with $AB$ as diameter is

The equation of tangent drawn from the origin to the circle ${x^2} + {y^2} - 2rx + 2hy + {h^2} = 0$ are

If $a_1 a_2=b_1 b_2$, then the lines $a_1 x+b_1 y+c_1=0$ and $a_2 x+b_2 y+c_2=0$ cut the coordinate axes in concyclic points. The equation of the circle, is

For the two circles ${x^2} + {y^2} = 16$ and ${x^2} + {y^2} - 2y = 0$ there is/are

Tangents are drawn from the point $(17,7)$ to the circle $x^2+y^2=169$
Statement I The tangents are mutually perpendicular. because

Statement II The locus of the points from which mutually perpendicular tangents can be drawn to the given circle is $x^2+y^2=338$