Circles Questions (554)

Let \(\tau\) be a circle with centre \(C(3, 5)\) and \(PA\) and \(PB\) are pair of tangents drawn from an external point \(P(9, 11)\) to the circle \(\tau\). Find the distance between the origin and the point inside the quadrilateral \(ACBP\) which is equidistant from its four vertices.

Consider a series of \(n\) concentric circles \(C_1, C_2, \ldots, C_n\) with radii \(r_1, r_2, r_3, \ldots, r_n\) respectively satisfying \(r_1 > r_2 > r_3 > \ldots > r_n\) and \(r_1 = 10\). The circles are such that the chord of contact of tangents from any point on \(C_i\) to \(C_{i+1}\) is a tangent to \(C_{i+2}\) where \(i = 1, 2, 3, \ldots\) Find the value of \(\lim_{n \to \infty} \sum_{r=1}^{n} r_i\), if the angle between the tangents from any point of \(C_1\) to \(C_2\) is \(60^\circ\).

Let \(x\), \(y\), \(z\) and \(t\) be real numbers such that \((x, y)\) lies on a circle having radius 3; \((z, t)\) lies on a circle having radius 2 and \(xt - yz = 6\). Find the greatest value of \(P = xz\).[Note: Both circles have centre at origin.]

The number of points on \(y = \tan^{-1} x\), \(\forall x \in (0, \pi)\), whose image in \(y = x\) is the centre of the circle with radius \(\dfrac{\pi}{2\sqrt{2}}\) units and which is at a minimum distance of \(\dfrac{\pi}{2\sqrt{2}}\) units from the circle.

\(r\) be radius of incircle of triangle formed by joining centres of \((x-a)^2+(y-b)^2=9\), \((x-a)^2+(y-b-7)^2=16\) and circle touching above two circles and having radius 5 units. Find \(r^2/4\).

If a circle C passing through the point (4, 0) touches the circle \(x^2 + y^2 + 4x - 6y = 12\) externally at the point (1, −1), then the radius of C is __________.

A circle is inscribed in an equilateral triangle of side of length 12. If the area and perimeter of any square inscribed in this circle are $m$ and $n$, respectively, then $m+n^2$ is equal to:

The number of integral values of \(\alpha\) for which the point \((\alpha - 1,\ \alpha + 1)\) lies in the larger segment of the circle \(x^2 + y^2 - x - y - 6 = 0\) made by the chord whose equation is \(x + y - 2 = 0\) is

Given line $L: 4x + 3y = \lambda$. Circle $S: x^2 + y^2 = 6x + 4y = 12$. Centre $(3, -2)$, Radius $= 5$. Now, the line will be tangent to the circle if $p = r$. $\frac{|4(3) + 3(-2) - \lambda|}{\sqrt{16 + 9}} = 5 \Rightarrow \lambda = 31, -10$. So, for only one point of intersection inequality, $4x + 3y \leq \lambda$ must satisfy the centre of the circle.

If the angle of intersection at a point where the two circles with radii 5 cm and 12 cm intersect is 90°, then the length (in cm) of their common chord is __________ (up to three decimal places).

Let $C_1$ be circle $x^2 + y^2 + 8x + 8y = 0$, $C_2$ be circle $x^2 + y^2 + 8x - 8y = 0$, $C_3$ be circle $x^2 + y^2 - 8x + 8y = 0$, and $C_4$ be circle $x^2 + y^2 - 8x - 8y = 0$. Now, common tangents between $C_1$ and $C_2$ is 2, common tangents between $C_1$ and $C_3$ is 2, common tangents between $C_1$ and $C_4$ is 3, common tangents between $C_2$ and $C_3$ is 3, common tangent between $C_3$ and $C_4$ is 2, and common tangent between $C_1$ and $C_4$ is 2.

The area of a circle is 154 (units)\(^2\). Find the radius of the circle (using \(\pi = \dfrac{22}{7}\)):

The point \((1, 4)\) is inside the circle \(S\) whose equation is of the form \(x^2 + y^2 - 6x - 10y + k = 0\), \(k\) being an arbitrary constant. Find the possible values of \(k\) if the circle \(S\) neither touches the axes nor cuts them.

A circle passing through origin O cuts two straight lines \(x - y = 0\) and \(x + y = 0\) in points A and B respectively. If abscissae of A and B are roots of the equation \(x^2 + ax + b = 0\), then the equation of the given circle is:

Given a line segment AB, A ≡ (0, 0) and B(a, 0). Three circles S₁, S₂, S₃ of radius R are centred at the end points and the midpoint of the line segment AB. A fourth circle S₄ is drawn touching the 3 given circles. If 0

A square is inscribed in the circle $x^2+y^2-10x-6y+30=0$. One side of this square is parallel to $y=x+3$. If $(x_i,y_i)$ are the vertices of the square, then $\sum(x_i^2+y_i^2)$ is equal to:

A circle passing through the point $P(\alpha,\beta)$ in the first quadrant touches the two coordinate axes at the points $A$ and $B$. The point $P$ is above the line $AB$. The point $Q$ on the line segment $AB$ is the foot of perpendicular from $P$ on $AB$. If $PQ$ is equal to $11$ units, then the value of $\alpha\beta$ is _______.

Let $O$ be the origin and $OP$ and $OQ$ be the tangents to the circle $x^2+y^2-6x+4y+8=0$ at the points $P$ and $Q$ on it. If the circumcircle of the triangle $OPQ$ passes through the point $\left(\alpha,\dfrac{1}{2}\right)$, then a value of $\alpha$ is

Let the centre of a circle $C$ be $(\alpha,\beta)$ and its radius $r<8$. Let $3x+4y=24$ and $3x-4y=32$ be two tangents and $4x+3y=1$ be a normal to $C$. Then $(\alpha-\beta+r)$ is equal to

Let the circles $C_1:(x-\alpha)^2+(y-\beta)^2=r_1^2$ and $C_2:(x-8)^2+\left(y-\dfrac{15}{2}\right)^2=r_2^2$ touch each other externally at the point $(6,6)$. If the point $(6,6)$ divides the line segment joining the centres of the circles $C_1$ and $C_2$ internally in the ratio $2:1$, then $(\alpha+\beta)+4(r_1^2+r_2^2)$ equals:

Consider two circles $C_1: x^2+y^2=25$ and $C_2:(x-\alpha)^2+y^2=16$, where $\alpha\in(5,9)$. Let the angle between the two radii (one to each circle) drawn from one of the intersection points of $C_1$ and $C_2$ be $\sin^{-1}\left(\frac{\sqrt{63}}{8}\right)$. If the length of common chord of $C_1$ and $C_2$ is $\beta$, then the value of $(\alpha\beta)^2$ equals

Let the line $x + y = 1$ meet the circle $x^2 + y^2 = 4$ at the points $A$ and $B$. If the line perpendicular to $AB$ and passing through the mid point of the chord $AB$ intersects the circle at $C$ and $D$, then the area of the quadrilateral $ADBC$ is equal to

Let a circle $C$ of radius 1 and closer to the origin be such that the lines passing through the point $(3,2)$ and parallel to the coordinate axes touch it. Then the shortest distance of the circle $C$ from the point $(5,5)$ is:

Let the centre of a circle, passing through the points $(0,0)$, $(1,0)$ and touching the circle $x^2+y^2=9$, be $(h,k)$. Then for all possible values of the coordinates of the centre $(h,k)$, $4(h^2+k^2)$ is equal to ________.

If the points of intersection of the ellipses $x^2+2y^2-6x-12y+23=0$ and $4x^2+2y^2-20x-12y+35=0$ lie on a circle of radius $r$ and centre $(a,b)$, then the value of $ab+18r^2$ is

Let $y=x$ be the equation of a chord of the circle $C_1$ (in the closed half-plane $x\geq0$) of diameter 10 passing through the origin. Let $C_2$ be another circle described on the given chord as its diameter. If the equation of the chord of the circle $C_2$, which passes through the point $(2,3)$ and is farthest from the center of $C_2$, is $x+ay+b=0$, then $a-b$ is equal to

A pair of perpendicular lines passing through P(1, 4) intersect x-axis at Q and R, then locus of incentre of \(\triangle\)PQR, is:

Two Circles of radii 36 units and 9 units touch each other externally, a third circle of radius $r$ touches the two given circles externally and also their common tangent, then the value of $r$ is:

Let $O$ be the centre of the circle $x^2+y^2=r^2$, where $r>\dfrac{5}{2}$. Suppose $PQ$ is a chord of this circle and the equation of the line passing through $P$ and $Q$ is $2x+4y=5$. If the centre of the circumcircle of $\triangle OPQ$ lies on the line $x+2y=4$, then the value of $r$ is:

Lines $L_1$, $L_2$ are parallel. $C_1$ is tangent to both. $C_2$ is tangent to $C_1$ externally and $L_1$; $C_3$ is tangent to $C_1$ externally and $L_2$; both between the lines and tangent to each other. Radii $r_2=5\sqrt{2}$, $r_3=10\sqrt{2}$. Diameter of $C_1$:

The line $y=2x$ touches a circle with centre $(0,\alpha)$, $\alpha>0$, radius $r$ at point $A_1$. $B_1$ is the diametrically opposite point. Given $\alpha+r=5+\sqrt{5}$. Match: P)$\alpha$; Q)$r$; R)$A_1$; S)$B_1$ List-II: 1)$(-2,4)$; 2)$\sqrt{5}$; 3)$(-2,6)$; 4)$\sqrt{5}$; 5)$(2,4)$

Let $O$ be the centre of the circle $x^2+y^2=r^2$, where $r>\dfrac{5}{2}$. Suppose $PQ$ is a chord of this circle and the equation of the line passing through $P$ and $Q$ is $2x+4y=5$. If the centre of the circumcircle of $\triangle OPQ$ lies on the line $x+2y=4$, then the value of $r$ is:

Lines $L_1$, $L_2$ are parallel. $C_1$ is tangent to both. $C_2$ is tangent to $C_1$ externally and $L_1$; $C_3$ is tangent to $C_1$ externally and $L_2$; both between the lines and tangent to each other. Radii $r_2=5\sqrt{2}$, $r_3=10\sqrt{2}$. Diameter of $C_1$:

The line $y=2x$ touches a circle with centre $(0,\alpha)$, $\alpha>0$, radius $r$ at point $A_1$. $B_1$ is the diametrically opposite point. Given $\alpha+r=5+\sqrt{5}$. Match: P)$\alpha$; Q)$r$; R)$A_1$; S)$B_1$ List-II: 1)$(-2,4)$; 2)$\sqrt{5}$; 3)$(-2,6)$; 4)$\sqrt{5}$; 5)$(2,4)$

ABCD is a cyclic quadrilateral, where AB, BC, CD are represented by x + y - 1 = 0, 2x + y + 1 = 0 and x - 3y + 5 = 0 respectively. If DA passes through (2, 1), then the equation of DA is

A circle S = 0 passes through points of intersection of circles x2 + y2 - 2x + 4y = 1 and x2 + y2 + 4x - 2y - 5 = 0 and cuts the circle x2 + y2 - 4 = 0 orthogonally. Then the length of tangent from origin on circle S = 0, is :

A circle is inscribed in a rhombus $ABCD$ with one angle $60°$. The distance from the centre of the circle to the nearest vertex is equal to $1$. If $P$ is any point on the circle, then $|PA|^2 + |PB|^2 + |PC|^2 + |PD|^2$ is equal to:

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:

What is the equation of the circle which passes through the point of intersection of the circles S1: x2 + y2 - 6x + 2y + 4 = 0, S2: x2 + y2 + 2x - 4y - 6 = 0 and whose centre lies on the line y = x. 

Two tangents to the circle x2 + y2 = 4 at the points A and B meet at point P(-4, 0). The area of the quadrilateral PAOB in sq. units, where O is origin, is

The circle passing through (1, - 2) and touching the axis of x at (3, 0) also passes through the point

As shown in the figure, three circles which have the same radius $r$ have centres at $(0,0)$, $(1,1)$, and $(2,1)$. If they have a common tangent line, as shown, then the value of $10\sqrt{5r}$ is ___.

If the $C_1, C_2, C_3$ and $C$ are four circles of radius $r_1, r_2, r_3, r$ respectively as shown in figure:

Let a circle $C$ pass through the points $(4, 2)$ and $(0, 2)$, and its centre lie on $3x + 2y + 2 = 0$. Then the length of the chord, of the circle $C$, whose mid-point is $(1, 2)$, is

The lengths of the tangents from any point on the circle $x^2 + y^2 + 8x + 1 = 0$ to the circles $x^2 + y^2 + 7x + 1 = 0$ and $x^2 + y^2 + 4x + 1 = 0$ are in the ratio

The set of all values of $a^2$ for which the line $x+y=0$ bisects two distinct chords drawn from a point $P\!\left(\dfrac{1+a}{2},\dfrac{1-a}{2}\right)$ on the circle $2x^2+2y^2-(1+a)x-(1-a)y=0$ is equal to:

Circle $x^2 + y^2 + 16x + 12y + c = 0$ is touched by a straight line with slope $2$ and $y$-intercept $5$ units at a point $Q$. Then the coordinates of $Q$ are

If the radius of the largest circle with centre $(2,0)$ inscribed in the ellipse $x^2+4y^2=36$ is $r$, then $12r^2$ is equal to

Let Q be a point from where tangents drawn to circle $g(x,y) = 0$ are mutually perpendicular. If A, B are the points of contact of tangent drawn from Q to circle $g(x,y) = 0$, then area of triangle QAB is:

The angle between the chords of the circle $x^2 + y^2 = 100$, which passes through the point $(7,1)$ and also divides the circumference of the circle into two arcs whose lengths are in the ratio $2 : 1$, is equal to