Circles practice · Medium

For all values of $a$, equation of a diameter of the circle $x^2 + y^2– 2ax+2ay+ a^2 = 0$ is

If the chord $y=m x+1$ of the circle $x^{2}+y^{2}=1$ subtends an angle of measure $45^{\circ}$ at the major segment of the circle then value of $m$ is

The point on the circle $x^2+y^2=a^2$ in first quadrant, so that tangent drawn at this point make a triangle of area $a^2$ with the coordinate axes, is

If $y=2 x$ is a chord of the circle $x^2+y^2-10 x=0$, then the equation of a circle with this chord as diameter, is $x^2+y^2-a x-b y=0$. Sum of $a$ and $b$ is

If a conic passes through $(1,0)$ and satisfies differential equation $\left(1+y^2\right) d x-x y d y=0$. Then the equation of circle which is touching this conic at $(\sqrt{2}, 1)$ and passing through one of its foci is:

If the chord of contact of tangents from a point on the circle ${x^2} + {y^2} = r_1^2$ to the circle ${x^2} + {y^2} = r_2^2$ touches the circle ${x^2} + {y^2} = r_3^2$, then ${r_,},{r_2},{r_3}$ are in

Consider the circles $x^2+y^2=9$ and $x^2+y^2-10 x+9=0$, then find the length of the common chord of circle is

If a line, $y = mx + c$ is a tangent to the circle, ${(x - 3)^2} + {y^2} = 1$ and it is perpendicular to a line ${L_1},$ where ${L_1}$ is the tangent to the circle, ${x^2} + {y^2} = 1$ at the point $\left( {\frac{1}{{\sqrt 2 }},\,\frac{1}{{\sqrt 2 }}} \right)$; then

The line joining $(5,0)$ to $(10 \cos \theta, 10 \sin \theta)$ is divided internally in the ratio $2: 3$ at $P$. If $\theta$ varies, then the locus of $P$ is

How many common tangents can be drawn to the following circles $x^2+y^2=6 x$ and $x^2+y^2+6 x+2 y+1=0?$

Let $O(0,\,0)$ and $A(0,\,1)$ be two fixed points, then the locus of a point $P$ such that the perimeter of $\Delta AOP$ is 4, is

Locus of mid point of the chords of contact of $x^2+y^2=2$ from the points on the line $3 x+4 y=10$ is a circle with centre $P$. If $O$ is the origin then $O P$ is equal to:

The locus of a point which moves such that the sum of the squares of its distance from three vertices of a triangle is constant is a/an

Four distinct points $(2 k, 3 k),(1,0),(0,1)$ and $(0,0)$ lie on a circle for $k$ equal to:

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

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 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 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

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 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 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 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

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 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

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 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 :

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