Circle PYQs - Last 10 Years

The equation of the circle which passes through the points $(2, 3)$ and $(4, 5)$ and whose centre lies on a straight line $4x - y - 3 = 0$, is

The equation of a circle whose center lies on $x + 2y = 0$ and touching the lines $3x - 4y + 8 = 0$ and $3x - 4y - 28 = 0$ is

A line making equal intercepts on coordinate axes and is tangent to the circle $x^2 + y^2 = 4$. The length of each intercept made by line on the coordinate axes is ...

A circle passes through the point $(0,1)$ and touches the parabola $y = x^2$ at the point $(1,1)$. The centre of the circle is...

The tangent to the circle $x^2 + y^2 = 10$ at the point $(3,1)$ touches the circle $x^2 + y^2 - 2\sqrt{10}\,x - 20y + k = 0$, then the value of $k$ is...

The area of the region (in sq. unit) bounded by x-axis, the tangent and normal to the circle $x^2 + y^2 = 4$, drawn at a point $(1, \sqrt{3})$ is

The number of circles passing through the origin and touching the lines $x + y = 1$ and $x - y = 1$ is $\ldots$

If a circle passes through the points $(2,3)$ and $(4,5)$ and its center lies on the straight line $y - 4x + 3 = 0$, then its equation is......

If the circles $x^2 + y^2 = 16$ and $x^2 + y^2 + 2ax + 4y + 4 = 0$ touch each other internally then $a =$

The equations of the tangent to the curve $x^2 + y^2 = 10$, where the tangent is parallel to the line $2x + y - 1 = 0$, are

Line $l : x + y = 4$ intersects the circle $x^2 + y^2 - 2x - 2y = 2$ at points $A$ and $B$. If C is the center of the circle, then the area of $\triangle ABC$ is...

The centre and radius of the circle $(a+1)x^2 + 3y^2 - 6x + 9y + a + 4 = 0$ are respectively ...

The angle between the tangents drawn from the origin to the circle $(x-7)^2 + (y+1)^2 = 25$ is

If a circle with center $(-1, 1)$ touches the line $x + 2y + 4 = 0$, then the co-ordinates of the point of contact are

The equation of the circle concentric with circle $x^2 + y^2 - 6x + 7 = 0$ and which touches the line $x + y + 3 = 0$ is ....

The equation of a circle which passes through the points $(2,3)$ and $(4,5)$ and whose center lies on the straight line $y - 4x + 3 = 0$ is

If the equation $3x^2 + (3 - p)xy + qy^2 - 2px = 8pq$ represents a circle, then the area (in sq. units) of this circle is

If one of the diameters of the circle, given by the equation $x^2+y^2-4 x+6 y-12=0$, is a chord of a circle, ' S ', whose centre is at $(-3,2)$, then the length of radius of ' S ' is _______ units.

The least distance of the point $\mathrm{A}(10,7)$ from the circle $x^2+y^2-4 x-2 y-20=0$ is length of seg AM . If $\mathrm{MM}^{\prime}$ is the diameter of the circle, then the lengths of AM and $\mathrm{AM}^{\prime}$ are respectively ____...

Two tangents to the circle $x^2+y^2=4$ at the points A and B meet at $\mathrm{P}(-4,0)$. Then the area of quadrilateral PAOB, where ' $O$ ' is the origin is

A pair of tangents are drawn to the circle $x^2+y^2+6 x-4 y-12=0$ from a point $\mathrm{P}(-4,-5)$, then the area enclosed between these tangents and the area of the circle is

If a circle with centre at $(-1,1)$ touches the line $x+2 y+4=0$ then the co-ordinates of the point of contact are

Let the circle with centre at origin pass through the vertices of an equilateral triangle ABC . If $A \equiv(2,4)$, then the length of the median through A is

The equations of the tangents to the circle $x^2+y^2=36$ which are perpendicular to the line $5 x+y-2=0$ are

If the tangent and the normal at the point $(\sqrt{3}, 1)$ to the circle $x^2+y^{2 }=4$, and the X -axis form a triangle, then the area (in sq.units) of this triangle is

The equations of the tangents to the circle $x^2+y^2=36$ which are perpendicular to the line $5 x+y=2$, are

The minimum distance and maximum distance of the point $\mathrm{P}(2,-7)$ from the circle $x^2+y^2-14 x-10 y-151=0$ are respectively _______units

The locus of point of intersection of the tangents to the circle $x^2+y^2=16$, such that the angle between them is $60^{\circ}$, is

The equation of the circle passing through the point $(1,1)$ and having two diameters along the pair of lines $x^2-y^2-2 x+4 y-3=0$ is

The number of integral values of $k$ for which $x^2+y^2+\mathrm{k} x+(1-\mathrm{k}) y+5=0$ represents a circle whose radius cannot exceeds 5 , are

The number of common tangents that can be drawn to the circles $x^2+y^2-6 x=0$ and $x^2+y^2+6 x+2 y+1=0$ is __________

If the tangent at $(1,7)$ to the curve $x^2=y-6$ touches the circle $x^2+y^2+16 x+12 y+\mathrm{C}=0$, then $\mathrm{C}=$

The equation of the circle, the end points of whose diameter are the centres of the circles $x^2+y^2+6 x-14 y+5=0$ and $x^2+y^2-4 x+10 y-4=0$ is

The equation of the circle, concentric with the circle $x^2+y^2-6 x-4 y-12=0$ and touching the $\mathrm{X}$-axis is

The equation of the circle which has its centre at the point $(3,4)$ and touches the line $5 x+12 y-11=0$ is

The equation of the circle which passes through the centre of the circle $x^2+y^2+8 x+10 y-7=0$ and concentric which the circle $2 x^2+2 y^2-8 x-12 y-9=0$ is

The equation of the concentric circle, with the circle $\mathrm{C}_1$ having equation $x^2+y^2-6 x-4 y-12=0$ and having double area compared to the area of $\mathrm{C}_1$, is

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

If $\left(m_i, \frac{1}{m_i}\right), m_i>0, i=1,2,3,4$ are four distinct points on a circle, then the product $\mathrm{m}_1 \mathrm{~m}_2 \mathrm{~m}_3 \mathrm{~m}_4$ is equal to

Two tangents drawn from $\mathrm{P}(1,7)$ to the circle $x^2+y^2=25$, touch the circle at Q and R respectively. The area of the quadrilateral PQOR is

The equation of the circle, concentric with the circle $2 x^2+2 y^2-6 x+8 y+1=0$ and double of its area is

One end of the diameter of the circle $x^2+y^2-6 x-5 y-1=0$ is $(-1,3)$, then the equation of the tangent at the other end of the diameter is

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

If the sides of a rectangle are given by the equations $x=-2, x=6, y=-2, y=5$, then the equation of the circle, drawn on the diagonal of this rectangle as its diameter, is

The equation of the tangent to the circle, given by $x=5 \cos \theta, y=5 \sin \theta$ at the point $\theta=\frac{\pi}{3}$ on it , is

The parametric equations of the circle $x^2+y^2-\mathrm{a} x-b y=0$ are

The abscissae of the two points A and B are the roots of the equation $x^2+2 a x-b^2=0$ and their ordinates are roots of the equation $y^2+2 p y-q^2=0$. Then the equation of the circle with AB as diameter is given by

Let PQ and RS be tangents at the extremities of the diameter PR of a circle of radius $r$. If PS and RQ intersect at a point X on the circumference of the circle, then 2 r equals

Two tangents to the circle \(x^2+y^2=4\) at the points \(\mathrm{A}\) and \(\mathrm{B}\) meet at the point \(\mathrm{P}(-4,0)\). Then the area of the quadrilateral \(\mathrm{PAOB}, \mathrm{O}\) being the origin, is

The sides of a rectangle are given by the equations \(x=-2, x=4, y=-2\) and \(y=5\)
Then the equation of the circle, whose centre is the point of intersection of the diagonals, lying within the rectangle and touching only two opposite sides...