Circles Questions (554)

Let $PQ$ and $RS$ be the 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 the value of $2r$ is equal to

Combined equation to the pair of tangents drawn from the origin to the circle \(x^2 + y^2 + 4x + 6y + 9 = 0\) is

In the xy plane, the segment with end points (3, 8) and (–5, 2) is the diameter of the circle. The point (k, 10) lies on the circle for

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 :

The locus of the mid points of the chords of the circle $C:(x-4)^2+(y-5)^2=4$ which subtend an angle $\theta_1$ at the centre of circle $C_1$, is a circle of radius $r_1$. If $\theta_1=\dfrac{\pi}{3}$, $\theta_3=\dfrac{2\pi}{3}$ and $r_1^2=r_2^2+r_3^2$, then $\theta_2$ is equal to:

If the tangents at the points $P$ and $Q$ on the circle $x^2+y^2-2x+y=\dfrac{5}{4}$ meet at the point $R\!\left(\dfrac{9}{4},2\right)$, then the area of the triangle $PQR$ is

Tangents are drawn at the point of intersections of the circles $x^2 + y^2 = 1$ and $x^2 + y^2 - (\lambda + 6)x + (8 - 2\lambda)y - 3 = 0$. ($\lambda$ being the variable). Then the locus of the point of intersection of these tangents is:

In a circle with centre $O$ $PA$ and $PB$ are two chords. $PC$ is the chord that bisects the angle $APB$. The tangent to the circle at $C$ is drawn meeting $PA$ and $PB$ extended at $Q$ and $R$ respectively. If $QC = 3$, $QA = 2$ and $RC = 4$, then length of $RB$ equals:

The area of region bounded by circle $f(x,y) = 0$ with x-axis in the first quadrant is:

The locus of the mid points of the chords of the circle $x^2 + y^2 - ax - by = 0$ which subtend a right angle at $(a/2, b/2)$ is:

Considering the circles, $x^2 + y^2 = 25$ and $x^2 + y^2 = 9$. From the point $A(0, 5)$ two segments are drawn touching the inner circle at the points $B$ and $C$ while intersecting the outer circle at the points $D$ and $E$. If $O$ is the centre of both the circles then the length of the segment $OF$ that is perpendicular to $DE$, is:

If the lengths of tangents from $P(1,3)$ and $Q(3,7)$ to a circle arc $\sqrt{2}$ units and $\sqrt{18}$ units respectively, then the length of the tangent from $R(7,15)$ to the same circle is

If the circles x2 + y2 + 2ax + 2by + c = 0 and x2 + y2 + 2bx + 2ay + c = 0 where c > 0, have exactly one point in common then the value of \(\frac{(a+b)^{2}}{2 c}\) is :

The locus of the center of the circles such that the point (2, 3) is the mid point of the chord 5x + 2y = 16 is

Let a circle of radius 4 pass through the origin O, the points $A(-\sqrt{3}a,0)$ and $B(0,-\sqrt{2}b)$, where $a$ and $b$ are real parameters and $ab\neq0$. Then the locus of the centroid of $\triangle OAB$ is a circle of radius

Let $ABCD$ be a quadrilateral in which $AB \parallel CD$, $AB \perp AD$ and $AB = 3CD$. If the area of the quadrilateral $ABCD$ is 4, then the radius of the circle touching all the four sides of the quadrilateral is:

Let the tangents at the points $A(4,-11)$ and $B(8,-5)$ on the circle $x^2+y^2-3x+10y-15=0$ intersect at the point $C$. Then the radius of the circle, whose centre is $C$ and the line joining $A$ and $B$ is its tangent, is equal to:

Let $C$ be a circle $x^2 + y^2 = 1$. The line $y = mx + m$ intersects $C$ at the point $P$ other than $(-1, 0)$, the number of rational choices for $m$ for which both the coordinates of $P$ are rational, is:

Let the tangent and normal at the point $(3\sqrt{3},1)$ on the ellipse $\dfrac{x^2}{36}+\dfrac{y^2}{4}=1$ meet the $y$-axis at the points $A$ and $B$ respectively. Let the circle $C$ be drawn taking $AB$ as a diameter and the line $x=2\sqrt{5}$ intersect $C$ at the points $P$ and $Q$. If the tangents at the points $P$ and $Q$ on the circle intersect at the point $(\alpha,\beta)$, then $\alpha^2-\beta^2$ is equal to

Let C be the circle of radius unity centred at the origin. If two positive numbers \(x_1\) and \(x_2\) are such that the line passing through \((x_1, -1)\) and \((x_2, 1)\) is tangent to C, then:

The normal to the circle x2 + y2 - 3x - 6y - 10 = 0 at the point (-3, 4) is

Consider the points P((2, 1)); Q((0, 0)); R((4, -3)) and the circle S : x^2 + y^2 - 5x + 2y - 5 = 0

Chord of the curve 4x2 + y2 - x + 4y = 0 which subtend a right angle at the origin pass through a fixed point whose co-ordinate are :

Find the equation of circle which pass through (5, 9) and center at (2, 5).

If the shortest distance of the parabola \(y^{2}=\) \(4 x\) from the centre of the circle \(x^{2}+y^{2}-4 x-\) \(16 y+64=0\) is \(d\), then \(d^{2}\) is equal to

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

If in a triangle, R and r are the circumradius and inradius respectively, then the H.M. of the exradii of the triangle is :

The circle x2 + y2 - 4x - 4y + 4 = 0 is inscribed in a triangle that has two of its sides along the coordinate axes. If the locus of the circumcentre of the triangle is x + y - xy + k\(\sqrt{x^{2}+y^{2}}\) = 0, then k =

If circular arcs \(\widehat{A C}\) and \(\widehat{B C}\) have centres at B (-\(\alpha\), 0) and A(\(\alpha\), 0) respectively and equation of circle which touches both arcs \(\widehat{A C}\) and \(\widehat{B C}\) and line AB is (x - a)2 + (y - b)2 = r2 r > 0, if length of arc BC = 8\(\pi\), the value of |\(\alpha\) + b + r - \(\alpha\)| equals:

If a circle passes through the point (0, 0), (a, 0), (0, b), then its centre is

If the tangent at (1, 7) to the curve x2 = y - 6 touches the circle x2 + y2 + 16x + 12y + c = 0, then the value of c is

If p and q represent the lengths of a direct common tangent and a transverse common tangent respectively to the circles \(\omega_1\) : x2 + y2 + 14x - 4y + 28 = 0 and \(\omega_2\) : x2 + y2 - 14x + 4y - 28 = 0, then \(\frac{p}{q}\) is

The equation of the circle which touches x-axis at (3, 0) and passes through (1, 4) is given by

If the coordinates of one end of the diameter of the circle x2 + y2 - 8x - 4y + c = 0 are (-3, 2), then the coordinates of other end are

The locus of the centres of the circles, which touch the circle, x2 + y2 = 1 externally, also touch the Y-axis and lie in the first quadrant, is

The equation of circle is \((x-0)^2 + (y-2)^2 + \lambda x = 0\) which passes through the points \((-2, 4)\). The centre of the circle is:

Let C be the circle with centre (0, 0) and radius 3 units. The equation of the locus of the mid points of the chords of the circle C that subtend an angle of \(2\pi/3\) at its centre is

Let \(S \equiv x^2 + y^2 - 4x - 2y - 20\). If \(P(10, 7)\) lies outside the circle \(S = 0\), the greatest distance of point \(P\) from \(S = 0\) is:

Given two circles \((x-1)^2 + (y-3)^2 = l^2\) and \(x^2 + y^2 - 8x + 2y + 8 = 0\). If the two circles intersect in two distinct points, find the range of \(l\) (i.e., \(r\)):

A circle passes through the points \((2, 3)\) and \((4, 5)\). If the centre lies on the line \(y - 4x + 3 = 0\), then its radius is equal to

If a tangent to the circle \(x^2 + y^2 = 1\) intersects the coordinate axes at distinct points P and Q, then the locus of the mid-point of PQ is:

The equation of the tangent to the circle \(x^2 + y^2 + 4x - 4y + 4 = 0\) which make equal intercepts on the positive coordinate axes, is

Let x + 2y - 5 + \(\lambda\) (x + 3y - 7) = 0 be a variable chord of the circle x2 + y2 - 4x - 6y + 11 = 0. If perpendiculars are drawn at the end points of these chords pass through a fixed point (a, b)a \(\ne\) 1 then (a + b) is equal to:

The normal drawn at P (-1, 2) on the circle x2 + y2 - 2x - 2y - 3 = 0 meets the circle at another point Q. Then the coordinates of Q are

If the locus of the point, whose distances from the point \((2,1)\) and \((1,3)\) are in the ratio \(5: 4\), is \(a x^{2}+b y^{2}+\)\(c x y+d x+e y+170=0\), then the value of \(a^{2}+2 b+3 c+4 d+e\) is equal to:

Point M moved along the circle \((x-4)^2 + (y-8)^2 = 20\). Then it broke away from it and moving along a tangent to the circle cuts the x-axis at the point \((-2, 0)\) the co-ordinate of the point on the circle at which the moving point broke away can be

The lines \(2x + 3y + 19 = 0\) and \(9x + 6y - 17 = 0\) cut the coordinate axes in concyclic points.State whether the statement is true or false.

A chord AB drawn from the point A(0, 3) on circle x2 + 4x +(y - 3)2 = 0 meets to M in such a way that AM = 2AB, then the locus of point M will be

a, b > 0. The length of the common chord of the circles (x - a)2 + y2 = a2 and x2 + (y - b)2 = b2 is

Let \(A B C D\) and \(A E F G\) be squares of side 4 and 2 units, respectively. The point \(E\) is on the line segment \(A B\) and the point \(F\) is on the diagonal \(A C\). Then the radius \(r\) of the circle passing through the point \(F\) and touching the line segments \(B C\) and \(C D\) satisfies: