Circles Questions - JEE

If Q, S are two points on the circle $x^2 + y^2 = 4$ such that the tangents QP, SR are parallel. If PS, QR intersect at T then $\left(\dfrac{QT}{PQ}\right)^2 + \left(\dfrac{ST}{RS}\right)^2 + PQ \cdot RS \neq$

Circle touches the x-axis at (3, 0) and making an intercept of length 8 on the y-axis. The equation of the circle is:

The equation of the circle described on the chord $3x + y + 5 = 0$ of the circle $x^2 + y^2 = 16$ as diameter is

The least and the greatest distances of the point (10, 7) from the circle $x^2 + y^2 - 4x - 2y - 20 = 0$ are

The centre of a circle passing through the points (0, 0), (1, 0) and touching the circle $x^2 + y^2 = 9$ is

How many ordered pair of integers $(a, b)$ satisfy all the following inequalities: \(a^2 + b^2

A circle $S$ whose radius is $1$ unit, touches the X-axis at point $A$. The centre $O$ of $S$ lies in the first quadrant. The tangent from the origin $O$ to the circle touches it at $T$ and a point $P$ lies on it such that the triangle $OAP$ is a right-angled triangle at $A$ and its perimeter is $8$ unit. The length of $QP$ is ___.

If a circle touches the hypotenuse of a right-angled triangle at its middle point and passes through the middle point of shorter side. If $a$ and $b$ $(a < b)$ be the length of the sides and the radius of the circle is $\frac{b}{ka}\sqrt{a^2 + b^2}$, then the value of $k$ is __________.

Two circles are given by \[x^2 + y^2 - 8 = 0\] and \[(x - a)^2 + y^2 - 8 = 0\]. The common chord is obtained by subtracting the equations: \[2ax - a^2 = 0\], which gives \[x = \frac{a}{2}\]. If the common chord subtends a right angle at the origin, find the value of a.

For any $\lambda \in \mathbb{R}$, the locus of $x^2 + y^2 - 2\lambda x - 2\lambda y + \lambda^2 = 0$ touches the line

The circles $x^2 + y^2 + 2ax + 2ny = 0$ and $x^2 + y^2 + 2bx + 2ny = 0$ touch each other if

Consider the circles $C_1 \equiv x^2 + y^2 - 2x - 4y - 4 = 0$ and $C_2 \equiv x^2 + y^2 + 2x + 4y + 4 = 0$ and the line $L \equiv x + 2y + 2 = 0$, then

The tangent to the circle $C_1 : x^2 + y^2 - 2x - 1 = 0$ at the point $(2, 1)$ cuts off a chord of length 4 from a circle $C_2$ whose centre is $(3, -2)$. The radius of $C_2$ is

The given circle equation is $x^2 + y^2 + 2x - 4y - 4 = 0$. The centre of the given circle is (–1, 2) and its radius is 3. Find the centre of the required circle.

The equation of a circle which touches the line $2x - y = 1$ at (1, 1) and also touches the line $2x + y = 4$ is

On the side $AC$ of an acute angled triangle $ABC$ a point $D$ is taken, such that $AD = 1, DC = 2$ and $BD$ is an altitude of $\triangle ABC$. A circle of radius $2$, which passes through points $A$ and $D$ and touches a circle at the point $D$ circumscribed about the $\triangle BDC$. If the area of $\triangle ABC$ is $A$ then the value of $\frac{1}{11}[A]$ is equal to __________. (Where $[.]$ represents G.I.F)

The circle passing through the intersection of the circles, $x^2 + y^2 - 6x = 0$ and $x^2 + y^2 - 4y = 0$, having its centre on the line, $2x - 3y + 12 = 0$, also passes through the point

The straight line $y = mx + c$ cuts the circle $x^2 + y^2 = a^2$ in real points if $a\sqrt{1 + m^2} > |c|$.State whether the statement is true or false.

The tangent from point P(4, 7) is drawn to a circle with centre O(0, 0). If C is the point of contact of the tangent, find PA · PB where A and B are the ends of the chord through P.Given: PA · PB = (PC)², and the centre of the circle is O(0,0). Find PA · PB.

Let point P(x1, y1) be any point on the circle (x1 - 3)2 + (y1 + 2)2 = 5r2. Find the area between two circles if the length of tangent drawn from point P(x1, y1) to the circle (x - 3)2 + (y + 2)2 = r2 is such that the area is kπ.

We have two straight lines $x - y = 1$ and $2x + y = 3$. The tangent to the circle at point $P(1, -1)$ (where the two lines intersect) with centre $C\left(\dfrac{4}{3}, \dfrac{1}{3}\right)$ is:

If a circle passes through the point $(a, b)$ and cuts the circle $x^2 + y^2 = 4$ orthogonally, then the locus of its centre is

If a circle of radius 2 unit touches the $y$-axis at the origin, $'O'$ and intersects the lines $y = (2-\sqrt{3})x$ and $y = -(2+\sqrt{3})x$ in the I and IV quadrants at $A$ and $B$ respectively, then area of $\triangle AOB$ (in square units) is:

The number of possible integral values of $m$ for which the circle $x^2 + y^2 = 4$ and $x^2 + y^2 - 6x - 8y + m^2 = 0$ have exactly two common tangents is __________.

If a circle passes through the point $(a, b)$ and cuts the circle $x^2 + y^2 = k^2$ orthogonally, the equation of the locus of its centre is

The tangent and the normal lines at the point $(\sqrt{3}, 1)$ to the circle x2 + y2 = 4 and the X-axis form a triangle. The area of this triangle (in square units) is

If the line x + 2by + 7 = 0 is a diameter of the circle x2 + y2 - 6x + 2y = 0, then b =

A circle with radius 1 has diameter AB. C lies on this circle such that $\frac{\widehat{A C}}{\widehat{B C}}$ = 4. $\overline{A C}$ divides the circle into two parts, and we will label the smaller part Region I. Similarly, $\overline{B C}$ also divides the circle into two parts, and we will denote the smaller one as Region II. The difference between the areas of Region I and II is : (where $\widehat{A C}$ represents are AC, $\overline{A C}$ represents chord AC)

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

29. If a chord of the circle $x^2 + y^2 - 4x - 2y - c = 0$ is trisected at the points $(1/3, 1/3)$ and $(8/3, 8/3)$, then the radius of the circle will be:

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:

Circle(s) touching x-axis at a distance 3 from the origin and having an intercept of length $2\sqrt{7}$ on y-axis is(are):

A point P moves such that the sum of the squares of its distances from the sides of a given square is a constant. Then point P moves on

Let A and B are two points both lying within a given circle S, and P be a point on circumference of S at which AB subtends the greatest angle.Statement-1: If $A \equiv (1, 1)$, $B \equiv (1, -1)$ and equation of S is $x^2 + y^2 = 4$ then P will be $(2, 0)$Statement-2: P will be the point where a circle passing through A and B touches the circle S.

Let ABCD be a square of side length 2 units. $C_2$ is the circle through vertices A, B, C, D and $C_1$ is the circle touching all the sides of the square ABCD. L is a line through A.If P is a point on $C_1$ and Q is another point on $C_2$, then $\frac{PA^2 + PB^2 + PC^2 + PD^2}{QA^2 + QB^2 + QC^2 + QD^2}$ is equal to:

If the tangent to the conic, $y - 6 = x^2$ at (2, 10) touches the circle, $x^2 + y^2 + 8x - 2y = k$ (for some fixed $k$) at a point $(\alpha, \beta)$, then $(\alpha, \beta)$ is

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

Find the locus of the middle points of chords of a given circle $x^2 + y^2 = a^2$ which subtend a right angle at the fixed point $(p, q)$.

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. The area of circle circumscribing $\triangle ABC$ is:

The locus of the centres of the circles, which touch the circle, $x^2 + y^2 = 1$ externally, also touch the y-axis and lie in the first quadrant, is:

Consider 3 equal circles of radius $r_1$ within a circle of radius $r_2$ each to touch the other two and the given circle.Statement-1: $\frac{r_1}{r_2} = \frac{\sqrt{3}}{\sqrt{3}+1}$Statement-2: Incentre of triangle formed by joining centres of 3 equal circles is same as centre of given circle.

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

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

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. Let FG and AH intersect at point K, then the length of AK is:

Let A = (0,0), B = (4,0) and on segment AB is given a point M. On the same side of AB, squares AMCD and BMFE are constructed above AB. The circumcircles S₁ and S₂ of two squares AMCD and BMFE respectively have centres P and Q, and intersect in M and another point N.The point of intersection of the lines FA and BC is:

A circle $C$ passes through the points of intersection of the parabola $y + 1 = (x - 4)^2$ and the $x$-axis. The length of tangent from origin to $C$ is:

Let the bisector of ∠A of △ABC meets BC in D and the circumcircle of △ABC in E. Analyze: AD is less than the G.M. (Geometric Mean) of AB and AC.

If S₁, S₂ and S₃ are three circles congruent to C₂ and touch both C₁ and C₂; then the area of triangle formed by joining centres of the circles S₁, S₂ and S₃ is (in square units)

Let A = (0, 0), B = (4, 0) and on segment AB is given a point M. On the same side of AB, squares AMCD and BMFE are constructed above AB. The circumcircles S₁ and S₂ of two squares AMCD and BMFE respectively have centres P and Q, and intersect in M and another point N.For all positions of M varying along the segment AB, the line MN passes through the fixed point R(a, b), then a + b = ?

Circles Questions (554)