Circles Questions - JEE

If a circle C, whose radius is 3, touches externally the circle, x2 + y2 + 2x − 4y − 4 = 0 at the point (2, 2), then the length of the intercept cut by this circle C, on the x-axis is equal to

Two circles with radii $r_1$ and $r_2$ touch each other externally at point A, with centres $C_1$ and $C_2$. A common tangent touches them at B and C respectively. In $\triangle AC_1C_2$,$\cos A = \dfrac{r_1^2 + r_2^2 - (C_1C_2)^2}{2r_1r_2} = \cos 60^\circ$$r_1^2 + r_2^2 - (C_1C_2)^2 = r_1r_2$ ...(1)and $(C_1C_2)^2 - (r_1 - r_2)^2 = 25$ ...(2)Find $r_1 r_2$.

A line drawn through the point $P(4, 7)$ cuts the circle $x^2 + y^2 = 9$ at the points $A$ and $B$, then $PA \cdot PB$ is equal to

Let $C$ be a circle with radius $\sqrt{10}$ units and centre at the origin. Let the line $x+y=2$ intersects the circle $C$ at the points $P$ and $Q$. Let $M N$ be a chord of $C$ of length 2 unit and slope -1. Then, a distance (in units) between the chord $P Q$ and the chord $M N$ is

AB is the diameter of a circle along the x-axis whose centre is the origin and $A = (a, 0)$. $P(a\cos\alpha, \sin\alpha)$ and $Q(a\cos\beta, \sin\beta)$ are points on the circle. The locus of the point of intersection of $AP$ and $BQ$ is

Let z = x + iy. Let S1 denote the interior of circle of radius 4 units, S3 denotes x > 0, and $S_2 = \operatorname{Im}\left(\frac{(x-1+i(y+\sqrt{3}))(1+i\sqrt{3})}{4}\right) > 0$. Find the area of the region common to S1, S2, and S3.

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 $\vec{c}$ and $\vec{d}$ be vectors such that $|\vec{c}+\vec{d}|=\sqrt{29}$ and $\vec{c}\times(2\hat{i}+3\hat{j}+4\hat{k})=(2\hat{i}+3\hat{j}+4\hat{k})\times\vec{d}$. If $\lambda_1,\lambda_2$ ($\lambda_1>\lambda_2$) are the possible values of $(\vec{c}+\vec{d})\cdot(-7\hat{i}+2\hat{j}+3\hat{k})$, then the equation $K^2x^2+(K^2-5K+\lambda_1)xy+\left(3K+\dfrac{\lambda_2}{2}\right)y^2-8x+12y+\lambda_2=0$ represents a circle, for K equal to:

Let ABCD be a quadrilateral with area 18, with side AB parallel to the side CD and $AB = 2CD$. Let AD be perpendicular to AB and CD. If a circle is drawn inside the quadrilateral ABCD touching all the sides, then its radius is:

(B) $x^2 + 18x + (y - 3)^2 = 0$

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

The minimum distance between the circle $x^2 + y^2 = 9$ and the curve $2x^2 + 10y^2 + 6xy = 1$ is:

A common tangent is drawn to the circle x2 + y2 = c2 and the parabola y2 = 4ax. If the angle which this tangent makes with the axis of x is π/4, then the relationship between a and c (a, c > 0) is:

If $O$ is the origin and $OP, OQ$ are distinct tangents to the circle $x^2 + y^2 + 2gx + 2fy + c = 0$, then the circumcentre of the triangle $OPQ$ is

Using coordinate geometry, assume endpoints of hypotenuse as $A(3,0)$ and $B(0,4)$, the third vertex is origin, such that the midpoint of the hypotenuse is $P\left(\frac{3}{2},2\right)$ and the midpoint of the shorter side is $Q\left(\frac{3}{2},0\right)$. Let centre be $C(h, k)$.

In a right triangle ABC, right angled at A, on the leg AC as diameter, a semicircle is described. The chord joining A with the point of intersection D of the hypotenuse and the semicircle, then the length AC equals to:

Consider the circle $x^2 + y^2 - 10x - 6y + 30 = 0$. Let O be the centre of the circle and tangent at $A(7, 3)$ and $B(5, 1)$ meet at C. Let S = 0 represents family of circles passing through A and B, then:

If $a > 2b > 0$ then the positive value of m for which $y = mx - b\sqrt{1 + m^2}$ is a common tangent to $x^2 + y^2 = b^2$ and $(x - a)^2 + y^2 = b^2$ is:

The set of all real values of $\lambda$ for which exactly two common tangents can be drawn to the circles $x^2 + y^2 - 4x - 4y + 6 = 0$ and $x^2 + y^2 - 10x - 10y + \lambda = 0$ is the interval:

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

Let $A B C$ be the triangle such that the equations of lines $A B$ and $A C$ be $3 y-x=2$ and $x+y=2$, respectively, and the points $B$ and $C$ lie on $x$-axis. If $P$ is the orthocentre of the triangle $A B C$, then the area of the triangle $P B C$ is equal to

Two circles $S_1: x^2 + y^2 - 16 = 0$ and $S_2: x^2 + y^2 - 6x - 8y - 8 = 0$ intersect at points $A$ and $B$. Find the coordinates of point $P$ on the common chord.

Two chords are drawn from the point P(h, k) on the circle x2 + y2 = hx + ky. If the y-axis divides both the chords in the ratio 2:3, then which of the following may be correct?

Lines $L_1$ & $L_2$ are rotating in an anticlockwise direction about the points $A(-2, 0)$ and $B(2, 0)$ respectively in such a way that the speed of angle of rotation of $L_1$ with respect to $L_2$ is double. Initially equation of both lines are $y = 0$. If the angle of rotation of the $L_2$ varies between $0$ to $\frac{\pi}{2}$, then the locus of the point of intersection $P$ of lines $L_1$ & $L_2$ is part of a circle whose radius is equal to

Let $A(0, 3)$ and $B(0, 12)$ be two vertices of a $\triangle ABC$ where $C = (x, 0)$. If the circumcircle of $\triangle ABC$ touches the $z$-axis, then the value of $\cos 2\theta$ is (where, $\theta$ is angle $ACB$)

From a point $P(3, 3)$ on the circle $x^2 + y^2 = 18$, two chords $PQ$ and $PR$ each of 2 units length are drawn on this circle. The value of $\cos(\angle QPR)$ is equal to

A line has equation $\frac{x}{a} + \frac{y}{b} = 1$. If P is the foot of the perpendicular drawn from the origin to this line, with coordinates $(x_1, y_1)$, and the distance from centre $(2, 4)$ to the line $3x - 4y - k = 0$ equals the radius $5$, find the value of $k$.

Given the equation of the circle is $x^2 + y^2 + 4x - 10y - 7 = 0$, find the minimum distance from the point $(4, -3)$ to any point on the circle.

The locus of the centers of the circles which cut the circles $x^2 + y^2 + 4x - 6y + 9 = 0$ and $x^2 + y^2 - 5x + 4y - 2 = 0$ orthogonally is:

Given three circles of radii $a$, $b$, $c$ \((a x-axis as a common tangent. Then:

In a △ABC; inscribed circle with centre I touches sides AB, AC and BC at D, E, F respectively. Let area of quadrilateral ADIE is 5 units and area of quadrilateral BFID is 10 units. Find the value of cos(C/2)/sin((A-B)/2).

The line $2x - y + 1 = 0$ is tangent to the circle at the point $(2, 5)$ and the centre of the circles lies on $x - 2y = 4$. The radius of the circle is:

The lines 3x - 4y + 4 = 0 and 6x - 8y - 7 = 0 are tangents to the same circle whose radius is r, then 4r is equal to ?

If from a moving point P on $x^2 + y^2 = 4$ tangents PA and PB are drawn to $x^2 + y^2 = a^2$, then the locus of the circumcentre of triangle PAB, is \((0

Given a line segment AB, where 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\), B\) and C\) are points in the xy-plane such that A(1, 2)\); B(5, 6)\) and AC = 3BC\). Then:

It is given that \[\text{Im}\left(\frac{iz-2}{z-i}\right)+1=0\] The radius of the circle represented by this equation is:

The distance between the chords of contact of tangents to the circle $x^2 + y^2 + 2gx + 2fy + c = 0$ from the origin and the point $(g, f)$ is:

Consider two circles of radii $r_1$ and $r_2$ passing through vertex $A$ of $\triangle ABC$ and touching side $BC$ at points $B$ and $C$ respectively. If $a = 5$ and $\angle A = 30°$, then $\sqrt{r_1 r_2}$ is equal to:

$AB$ is tangent to the circle whose equation is $x^2 + y^2 = 9$. The coordinates of point $A$ are $(-10, 0)$ and point $B(a, b)$ is in the third quadrant. The slope of $AB$ is:

A circle touching the x-axis at (3, 0) and making an intercept of length 8 on the y-axis passes through the point

The equation(s) of the tangent at the point (0, 0) to the circle, making intercepts of lengths 2a and 2b units on the coordinates axes, is/are:

The radius of the circle passing through the point (6, 2) and having $x + y = 6$ as a normal and $x + 2y = 4$ as a diameter is

If the line x cos$\theta$ + y sin$\theta$ = 2, $\theta \ \in$ [0, 2$\pi$] touches the circle x2 + y2 - 2x = 0 then the number of such tangent(s), is:

Let P be a point on the line segment joining A(5cos α, 5sin α) and B(5cos β, 5sin β) such that 3PA = 2PB, then the locus of P is:

The sum of the squares of the lengths of the chords intercepted on the circle, $x^2 + y^2 = 16$, by the lines, $x + y = n$, $n \in N$, where $N$ is the set of all natural numbers, is __________.

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.

Circles Questions (554)