Circles Questions - JEE

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

A circle passing through the origin and cutting equal chords of length $\sqrt{2}$ from the straight line $y = x$ and $y = -x$...

A circle is inscribed in an equilateral triangle with side lengths 6 units. Another circle is drawn inside the triangle (but outside the first circle), tangent to the first circle and two of the sides of the triangle. The radius of the smaller circle is:

Consider a series of 'n' concentric circles $C_1, C_2, C_3, ..., C_n$ with radii $r_1, r_2, r_3, ..., r_n$ respectively, such that $n > r_2, ..., > r_n$ and $\sum r_i = 20$. If the tangents drawn from any point on $C_{i+1}$ are such that the chord of contact is a tangent to $C_{i+2}$ $(i = 1, 2, 3, ...)$ and the angle between the tangents from any point on $C_1$ to $C_2$ is $\frac{\pi}{3}$, then find the values of $\lim_{n \to \infty} \sum_{i=1}^{n} r_i$.

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:

Circles Questions (554)