Circles practice · Mixed difficulty

If the point $\left( { - 5 + \frac{\lambda }{{\sqrt 2 }}, - 3 + \frac{\lambda }{{\sqrt 2 }}} \right)$ is an interior point of the larger segment of the circle ${x^2} + {y^2} = 16$ cut off by the line $x + y = 2,$ then

Suppose $f(x, y)=0$ is the equation of a circle such that $f(x, 1)=0$ has equal roots (each equal to 2) and $f(1, x)=0$ also has equal roots (each equal to zero). The equation of circle is

An equation of the circle described on the chord $x \cos \alpha+y \sin \alpha-p=0$ of the circle $x^2+y^2=a^2$ as diameter is

A tangent is drawn to the circle $2\left( {{x^2} + {y^2}} \right) - 3x + 4y = 0$ and it touches the circle at point $A$. If the tangent passes through the point $P\left( {2,1} \right)$ then $PA =$

The smallest circle with centre on y-axis and passing through the point(7,3)has radius

Consider four circles $(x \pm 1)^2+(y \pm 1)^2=1$. Equation of smaller circle touching these four circles is

Consider a family of circles, which are passing through the point $\left( { - 1,\;1} \right)$ and are tangent to $x$-axis. If $\left( {h,\;k} \right)$ are the coordinates of the centre of the circles, then the set of values of $k$ is given by the interval

The circle $x^2+y^2=4 x+8 y+5$ intersects the line $3 x-4 y=m$ at two distinct points, if

The area of a circle and the area of a regular polygon of $n$ sides and of perimeter equal to that of the circle are in the ratio

A cone is made from a circular sheet of radius $\sqrt{3}$ by cutting out a sector and gluing the cut edges of the remaining piece together. The maximum volume attainable for the cone is

Equation of a circle with centre $(-4,3)$ touching internally and containing the circle $x^2+y^2=1$ is

The number of common tangents to the circles $x^2+y^2=4$ and $x^2+y^2-6 x-8 y=24$, is

A circle is passing through the points $A(1,1)$ and $B(1,3)$ and the bisector of first and third quadrant is normal to it, then its area, in square units, is

If tangents $P A$ and $P B$ are drawn to $x^2+y^2=9$ from the point $P(4,4)$, then area of $\triangle O A B, O$ being the origin, is equal to

The angle between the pair of tangents from the point $\left(1, \frac{1}{2}\right)$ to the circle $x^2+y^2+4 x+2 y-4=0$, is

The circles $x^2+y^2-10 x+16=0$ and $x^2+y^2=r^2$ intersect each other in two distinct points if

Find the intercept made by the circle $x^2+y^2-14 x-10 y+23=0$ on the line $7 x+y-4=0$.

The locus of pole of any tangent to the circle $x^2+y^2$ $=4$ w.r.t. the hyperbola $x^2-y^2=4$ is the circle

If the circles ${x^2} + {y^2} + 2gx + 2fy = 0$ and ${x^2} + {y^2} + 2g'\,x + 2f'\,y = 0$ touch each other, then

The range of values of $a$ such that the angle $\theta$ between the pair of tangents drawn from $\left( {a,\;0} \right)$ to the circle ${x^2} + {y^2} = 1$ satisfies $\frac{\pi }{2} < \theta < \pi$, is

If $y+3 x=0$ is the equation of a chord of the circle, $x^{2}+y^{2}-30 x=0$, then the equation of the circle with this chord as diameter is :

The circle described on the line joining the points $(0,1),(a, b)$ as diameter cuts the $x$-axis in points whose abscissae are roots of the equation

Two concentric circles of which smallest is $x^2+y^2=4$, have the difference in radii as $d$, if line $y=x+1$ cuts the circles in real points, then $d$ lies in the interval

The tangents to the circle $x^2+y^2=48$, which are inclined at angle of $60^{\circ}$ with the axis of $x$ form a rhombus, the length of whose sides is

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

If the chords of contact of the tangents from a point on the circle ${x^2} + {y^2} = {a^2}$ to the circle ${x^2} + {y^2} = {b^2}$ touch the circle ${x^2} + {y^2} = {c^2},$ then the roots of the equation $a{x^2} + 2bx + c = 0$, are

A line through $P\left( {1,4} \right)$ intersect a circle ${x^2} + {y^2} = 16$ at $A$ and $B$, then $PA.PB$ is equal to

Tangent to circle $x^2+y^2=5$ at $(1,-2)$, also touches the circle $x^2+y^2-8 x+6 y+20=0$. The coordinates of the corresponding point of contact is

If the base of a triangle and the ratio of the lengths of the other two unequal sides are given, then the vertex lies on a/an

The value of $c$, for which the line $y = 2x + c$ is a tangent to the circle ${x^2} + {y^2} = 16$, is