Circles Questions - JEE

Let $C: x^2+y^2=4$ and $C': x^2+y^2-4\lambda x+9=0$ be two circles. If the set of all values of $\lambda$ so that the circles $C$ and $C'$ intersect at two distinct points, is $\mathbb{R}-[a,b]$, then the point $(8a+12, 16b-20)$ lies on the curve:

If one of the diameters of the circle $x^2+y^2-10x+4y+13=0$ is a chord of another circle $C$, whose center is the point of intersection of the lines $2x+3y=12$ and $3x-2y=5$, then the radius of the circle $C$ 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 angle at which the circles $(x-1)^2 + y^2 = 10$ and $x^2 + (y-2)^2 = 5$ intersect is:

A circle of radius unity is centred at origin. Two particles start moving at the same time from the point $(1, 0)$ and move around the circle in opposite direction. One of the particle moves counter clockwise with constant speed u and the other moves clockwise with constant speed 3u. After leaving $(1, 0)$, the two particles meet first at a point P, and continue until they meet next at point Q. The coordinates of the point Q are:

An altitude BD and a bisector BE are drawn in the triangle ABC from the vertex B. It is known that the length of side AC = 1, and the magnitudes of the angles $\angle BEC$, $\angle ABD$, $\angle ABE$, $\angle BAC$ form an arithmetic progression.Let 'O' be the circumcentre of $\triangle ABC$, the radius of circle inscribed in $\triangle BOC$ 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, then the length of DE is:

A square $OABC$ is formed by line pairs $xy = 0$ and $xy + 1 = x + y$ where $O$ is the origin. A circle with centre $C_1$ inside the square is drawn to touch the line pair $xy = 0$ and another circle with centre $C_2$ and radius twice that of $C_1$, is drawn to touch the circle $C_1$ and the other line pair. The radius of the circle with centre $C_1$ is:

In an acute triangle ABC, point H is the intersection point of altitude CE to AB and altitude BD to AC. A circle with DE as its diameter intersects AB and AC at points F and G respectively. If BC = 25, BD = 20 and BE = 7.Area of the circle S is:

A common tangent of the two circles is:

The circle 'S' touches the sides AB and AD of the rectangle ABCD and cuts the side DC at a single point F and the side BC at a single point E. If |AB| = 32, |AD| = 40 and |BE| = 1.The angle between pair of tangents drawn from the point D to the circle 'S' is:

The circle 'S' touches the sides AB and AD of the rectangle ABCD and cuts the side DC at a single point F and the side BC at a single point E. If |AB| = 32, |AD| = 40 and |BE| = 1.The radius of circle is:

Let P is situated at a distance 'd' from centre O, then which of the following does not equal the product (PQ)(PR), where T is a point on C and PT is tangent to C:

Let C be a circle of radius r with centre at O, let P be a point outside C and D be a point on C. A line through P intersects C at Q and R, S is the midpoint of QR. For different choices of lines through P, what is the curve on which S lies:

The value of a + b + R equals, where P(a, b) is the centre and R is the radius of circle 'S' passing through points A(9, 3), B(7, –1) and C(1, –1).

If the circles $x^2 + y^2 + 5kx + 2y + k = 0$ and $2(x^2 + y^2) + 2kx + 3y - 1 = 0$, $(k \in R)$, intersect at the points P and Q, then the line $4x + 5y - k = 0$ passes through P and Q, for

Find the number of ordered pairs (a, b) that satisfy all three circle inequalities with centers at (1, 1), (1, 2), and (2, 1).

A parallelogram has vertices at A(0, 0), C(4, 4), and the midpoint of diagonal AC is E(2, 2). The locus of the extremities of the other diagonal is:

Let C be a circle of radius r with centre at O, let P be a point outside C and D be a point on C. A line through P intersects C at Q and R, S is the midpoint of QR. For different choices of lines through P, what is the curve on which S lies:

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. If \(0

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

Circles of radii 36 and 9 touch externally. The radius of the circle which touches the two circles externally and also their common tangent is:

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

A common tangent of the two circles is:

The circle S touches the sides AB and AD of the rectangle ABCD and cuts the side DC at a single point F and the side BC at a single point E. If $|AB| = 32$, $|AD| = 40$ and $|BE| = 1$. The area of trapezoid AFCB is:

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:

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

Question 3: Position of point $\left(\frac{1}{5}, 5A\right)$ with respect to circle is:

The radius of the circle, having centre at $(2, 1)$, whose one of the chord is a diameter of the circle $x^2 + y^2 - 2x - 6y + 6 = 0$

Find the equation of the circle circumscribing triangle DAO with diameter AE, where A(a,0) and E\left(0, \frac{3a}{4}\right).

Let the orthocentre and centroid of a triangle be $A(-3, 5)$ and $B(3, 3)$, respectively. If $C$ is the circumcentre of this triangle, then the radius of the circle having line segment $AC$ as diameter, is

The number of points of intersection of curve $\sin x = \cos y$ and circle $x^2 + y^2 = 1$.

Let C be the circle with centre at (1, 1) and radius = 1. If T is the circle centred at (0, y), passing through origin and touching the circle C externally, then the radius of T is equal to

The sum of the radii of inscribed and circumscribed circles for an $n$ sided regular polygon of side $a$, is

The two circles $x^2 + y^2 = ax$ and $x^2 + y^2 = c^2$ (c > 0) touch each other, if

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

Let $H$ and $L$ be a hyperbola and a line respectively. Solving $H$ and $L$ gives $2x^2 + 2x - 1 = 0$ with roots $\alpha$ and $\beta$. The equation of the family of circles passing through points $A$ and $B$ (intersections of $H$ and $L$) is $S_D + \lambda L = 0$. If this circle finally passes through $A(\alpha, -\alpha - 1)$, find the value of $\lambda - 1$ (i.e., the integer answer).

For the two circles $x^2 + y^2 = 16$ and $x^2 + y^2 - 2y = 0$, there is/are

The equation of the circle passing through the foci of the ellipse $\dfrac{x^2}{16} + \dfrac{y^2}{9} = 1$, and having centre at (0, 3) is

Centres of the three circles $x^{2} + y^{2} - 4x - 6y - 14 = 0$, $x^{2} + y^{2} + 2x + 4y - 5 = 0$ and $x^{2} + y^{2} - 10x - 16y + 7 = 0$

The centres of the circles $x^2 + y^2 = 1$, $x^2 + y^2 + 6x - 2y - 1 = 0$ and $x^2 + y^2 - 12x + 4y = 1$ are

If the circles $x^2 + y^2 - 16x - 20y + 164 = r^2$ and $(x - 4)^2 + (y - 7)^2 = 36$ intersect at two distinct points, then:

Tangents are drawn to a unit circle with centre at the origin from each point on the line 2x + y = 4. Then the equation to the locus of the middle point of the chord of contact is -

Let the tangents drawn to the circle, $x^2 + y^2 = 16$ from the point $P(0, h)$ meet the $x$-axis at points $A$ and $B$. If the area of $\triangle APB$ is minimum, then $h$ is equal to

Find the locus of the feet of the perpendiculars from the point $(h, k)$ to the tangents to the circle $x^2 + y^2 = 2ax$.

The circle for which the line joining the points $(am^2, 2am)$ and $\left(\dfrac{a}{m^2}, -\dfrac{2a}{m}\right)$ is a diameter, is

The length of the common chord of two circles of radii 3 and 4 unit which intersect orthogonally is $\frac{k}{5}$, then k equals ............

Three circles of radii a, b, c $(a < b < c)$ touch each other externally. If they have x-axis as a common tangent, then:

The line $x + 3y = 0$ is a diameter of the circle $x^2 + y^2 - 6x + 2y = 0$.State whether the statement is true or false.

The number of common tangents to the circles $x^2 + y^2 - 4x - 6y - 12 = 0$ and $x^2 + y^2 + 6x + 18y + 26 = 0$, is

Circles Questions (554)