Circles Questions (554)

Let $P(a_1,b_1)$ and $Q(a_2,b_2)$ be two distinct points on a circle with centre $C(\sqrt{2},\sqrt{3})$. Let $O$ be the origin and $OC$ be perpendicular to both $CP$ and $CQ$. If the area of the triangle $OCP$ is $\dfrac{\sqrt{35}}{2}$, then $a_1^2+a_2^2+b_1^2+b_2^2$ is equal to ___.

Let the circle $C_1: x^2+y^2-2(x+y)+1=0$ and $C_2$ be a circle having centre at $(-1,0)$ and radius 2. If the line of the common chord of $C_1$ and $C_2$ intersects the $y$-axis at the point $P$, then the square of the distance of $P$ from the centre of $C_1$ is:

If $(\alpha, \beta)$ is a point on a circle whose centre is on $x$-axis and has the coordinate $(\gamma, 0)$ which also touches the line $x + y = 0$ at $(2, -2)$, then:

Length of the tangents from the point \((1, 2)\) to the circles \(x^2 + y^2 + x + y - 4 = 0\) and \(3x^2 + 3y^2 - x - y - k = 0\) are in the ratio \(4:3\), then \(k\) is equal to

Points $P(-3,2)$, $Q(9,10)$ and $R(\alpha,4)$ lie on a circle $C$ with PR as its diameter. The tangents to $C$ at the points $Q$ and $R$ intersect at the point $S$. If $S$ lies on the line $2x-ky=1$, then $k$ is equal to ___.

Let the circle $x^2+y^2=4$ intersect the $x$-axis at the points $A(a,0)$, $a>0$ and $B(b,0)$. Let $P(2\cos\alpha,2\sin\alpha)$, $0<\alpha<\dfrac{\pi}{2}$ and $Q(2\cos\beta,2\sin\beta)$ be two points such that $(\alpha-\beta)=\dfrac{\pi}{2}$. Then the point of intersection of $AQ$ and $BP$ lies on:

The points of intersection of the line $ax+by=0$, $(a\neq b)$ and the circle $x^2+y^2-2x=0$ are $A(\alpha,0)$ and $B(1,\beta)$. The image of the circle with AB as a diameter in the line $x+y+2=0$ is:

The area of circle circumscribing $x = \frac{-3h}{2}$ is:

The number of points with positive integral coordinates satisfying $f(x,y) > 0, g(x,y) 3$ and $x < 6$ is:

The largest value of \(\dfrac{y}{x}\), where \((x, y)\) is a real number pair satisfying \((x-3)^2 + (y-3)^2 = 6\), is:

Let $PT$ be a tangent from the point $P(5,3 + \sqrt{3})$ to the circle $x^2 + y^2 + 4x - 6y - 3 = 0$, with centre $C$, at $T$ and $AB$ is a secant which passes through $P$ such that $BT$ is the normal at $T$. If $Ar(\triangle CAB) + Ar(\triangle CAT) = \frac{k}{25}$, then find the value of $(\sqrt{k} - 15)$ ([.] denotes G.I.F.).

Points $P$ and $Q$ are $3$ units apart. A circle centered at $P$ with a radius of $3$ units intersects a circle centered at $Q$ with radius $\sqrt{3}$ units at point $A$ and $B$. The area of the quadrilateral $APBQ$ is:

In the diagram, $DC$ is a diameter of the large circle centered at $A$, and $AC$ is a diameter of the smaller circle centered at $B$. If $DE$ is tangent to the smaller circle at $F$ and $DC = 12$ units then the length of $DE$ is:

If $A(0,a)$ and $B(0,\beta)$, $a, \beta > 0$ are two vertices of a variable triangle $ABC$, where the vertex $C(x, 0)$ is variable. The value of $x$ for which $\angle ACB$ is maximum is:

The value of $x_2$ is:

Two tangents are drawn from a point $P$ to the circle $x^2 + y^2 = 1$. If the tangents make an intercept of 2 units on the line $x = 1$, then locus of $P$ is:

(D) 18

One circle has a radius of 5 units and its center is at (0, 5). A second circle has a radius of 12 and its centre is at (12, 0). The radius of a third circle which passes through the center of the second circle and both points of intersection of the first two circles, is equal to:

Let k be an integer such that the triangle with vertices (k, - 3k), (5, k) and (- k, 2) has area 28 sq units. Then, the orthocentre of this triangle is at the point

If $r_1$ and $r_2$ are the radius of two circles passing through $(-1, 1)$ and touching the lines $x + y = 2, x - y = 2$, and $r_1 + r_2 = a\sqrt{2}$, then $a$ is equal to __________.

$CD$ is the common chord of the two circles of equal radii touching a line $L$ at $A$ and $B$. Let $C$ be closer to the line $L$ than $D$. The ratio of the radii of circumcircles of the triangle $AC$ and $ADB$ is ___.

Let $C_1, C_2, C_3$ be the centres of circles $S_1, S_2, S_3$ respectively, then which of the following must be true:

The circle $x^2 + y^2 + 6x - 24y + 72 = 0$ and $x^2 - y^2 + 6x + 16y - 46 = 0$ intersect at four points. The sum of distances from these four points to the point $(-3, 2)$ is $10k$. Then value of $k$ is equal to ___.

If the circles $x^2 + y^2 + (3 + \sin \beta)x + (2\cos \alpha)y = 0$ and $x^2 + y^2 + (2\cos \alpha)x + 2cy = 0$ touch each other, then the maximum value of $c$ is ___.

If circles \(x^2 + y^2 = c\) with radius 3 and \(x^2 + y^2 + ax + by + c = 0\) with radius 6 intersect at two points A and B. If length of AB = l. Find l.

Two tangents are drawn from a point $P$ to the circle $x^2 + y^2 = 1$. If the tangents make an intercept of 2 units on the line $x = 1$, then locus of $P$ is:

715. 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.]

If \(S_1: x^2 + y^2 = 4\) and \(S_2: x^2 + y^2 - 2ax - 2by + 2 = 0\) touches each other, then 4 times the radius of the circle \(S_2\) is:

The equation of the centre of a circle which touches the circle \(x^2 + y^2 - 6x + 14 = 0\) externally and also touches the y-axis is given by the equation

If the area of the quadrilateral formed by the tangents from the origin to the circle \(x^2 + y^2 + 6x - 10y + c = 0\) and the radii corresponding to the points of contact is 15, then a value of c is:

Let a circle $C_1$ be obtained on rolling the circle $x^2+y^2-4x-6y+11=0$ upwards 4 units on the tangent $T$ at $(3,2)$. Let $C_2$ be the image of $C_1$ in $T$. Let $A,B$ be the centres of $C_1,C_2$ and $M,N$ be the feet of perpendiculars from $A,B$ on the x-axis. Then the area of the trapezium AMNB is:

The shortest distance from the line $3x + 4y = 25$ to the circle $x^2 + y^2 = 6x - 8y$ is equal to:

If the two circles $(x - 1)^2 + (y - 3)^2 = r^2$ ($r > 0$) and $x^2 + y^2 - 8x + 2y + 8 = 0$ intersect in two distinct points, then the number of odd positive integral values of $r$ is ____.

Let a given line \(L_1\) intersect the \(x\) and \(y\) axes at P and Q respectively. Let another line \(L_2\) perpendicular to \(L_1\) cut the axes at R and S respectively. Find the locus of the point of intersection of the lines PS and QR.

If the circle \(C_1: x^2 + y^2 = 16\) intersects another circle \(C_2\) of radius 5 in such a manner that the common chord is of maximum length and has a slope equal to \(\frac{3}{4}\), then the coordinates of the centre of \(C_2\) are:

The radius of the circle passing through the point (6, 2), two of whose diameters are x + y = 6 and x + 2y = 4, is

If exactly two real common tangents can be drawn to the circles \(x^2 + y^2 - 2x - 2y = 0\) and \(x^2 + y^2 - 8x - 8y + \lambda = 0\) then

The $\lim_{n \to \infty} \frac{x_n}{2^n}$ is:

If the two circles $C_1: x^2 + y^2 = 16$ and circle $C_2$ of radius $5$ units intersect in such a manner that the common chord of maximum length has a slope equal to $3/4$, then the coordinates of the centre of $C_2$ are:

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:

For the circle x2 + y2 + 6x - 8y + 9 = 0, which of the following statement is true?

There are two real tangents from the point (2, 1) to the circle \(x^2 + y^2 = 8\).State whether the statement is true or false.

The straight-line x + 2y = 1 meets the coordinate axes at A and B. A circle is drawn through A, B and the origin. Then, the sum of perpendicular distances from A and B on the tangent to the circle at the origin is

Let the line segment joining the centres of the circles x2 - 2x + y2 = 0 and x2 + y2 + 4x + 8y + 16 = 0 intersect the circles at P and Q respectively. Then the equation of the circle with PQ as its diameter is

If A(a, 0) and B(−a, 0) are two fixed points and a point P moves such that ∠APB = 90°, then locus of P is

Let A = (-1, 0) and D = (0, -1). Two points B and C are such that points A, B, C, D are concyclic. Given that AB and CD are parallel and AB has equation x - y + 1 = 0. If B = (p, 2) and C = (r, 5), then r - s - p is

The radius of a circle, having minimum area, which touches the curve \(y = 4 - x^2\) and the lines \(y = |x|\) is

Let $S_1$ and $S_2$ denote the circles $x^2 + y^2 + 10x - 24y - 87 = 0$ and $x^2 + y^2 - 10x - 24y + 153 = 0$ respectively. (Let $m$ be the smallest positive value of $'a'$ for which the line $y = ax$ contains the centre of a circle which touches $S_2$ externally and $S_1$ internally). Given that $m^2 = \frac{p}{q}$, where $p$ and $q$ are relatively prime integers, if $(p + q)$ is equal to $13^k$, then the value of $k$ is equal to ______.

A variable circle passes through the point \(A(a, b)\) & touches the x-axis. Show that the locus of the other end of the diameter through A is \((x - a)^{2} = 4by\).

If the tangent at \((1, 7)\) to the curve \(x^2 = y - 6\) touches the circle \(x^2 + y^2 + 16x + 12y + c = 0\), then the value of \(c\) is