Circles Questions - JEE

ABCD is a square, the length of whose side is a. Taking AB and AD as the coordinate axes, the equation of the circle passing through the vertices of the square is

The area (in sq. units) of the part of the circle x2 + y2 = 36, which is outside the parabola y2 = 9x, is:

Let L = 0 be a common normal to the circle $x^2 + y^2 - 2\alpha x - 36 = 0$ and the curve $S : (1+x)^y + e^{xy} = y$ drawn at a point $x = 0$ on S, then the radius of the circle is

The centre of circle $C_2$ is:

If P is a point on the circle $x^2+y^2=4$, Q is a point on the straight line $5x+y+2=0$ and $x-y+1=0$ is the perpendicular bisector of PQ, then 13 times the sum of abscissa of all such points P is _____

If the point (1, 4) lies inside the circle $x^2 + y^2 - 6x - 10y + p = 0$ and the circle does not touch or intersect the coordinate axes, then the set of all possible values of $p$ is the interval:

Let A be the centre of the circle whose equation is x² + y² – 2x – 4y – 20 = 0. Suppose that the tangents at the points B(1, 7) and D(4, –2) on the circle meet at the point C. Find the area of the quadrilateral ABCD.

The equation of circle is $x^2 + y^2 + 4x - 4y + 4 = 0$. The equation of the tangent to this circle which makes equal intercepts on the positive coordinate axes is:

The centre of the circle given by $\vec{r} \cdot (\hat{i} + 2\hat{j} + 2\hat{k}) = 15$ and $|\vec{r} - (\hat{j} + 2\hat{k})| = 4$ is

Find the locus of the middle points of the chords of the circle $x^2 + y^2 = a^2$ which pass through a given point $(x_0, y_0)$.

A circle with centre C(3, 5) has AB as diameter, and P(9, 11) is a point outside the circle such that PA and PB are tangents to the circle. M(6, 8) is the midpoint of PC. The point inside the quadrilateral ACBP which is equidistant from all four vertices is M(6, 8). Find the distance from the origin to the point M.

The equation of a circle is $S_1 \equiv x^2 + y^2 = 1$. The orthogonal tangents to $S_1$ meet at another circle $S_2$ and the orthogonal tangents to $S_2$ meet at the third circle $S_3$. Then

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 equation of the tangent to the circle $x^2 + y^2 - 2x - 1 = 0$ at the point (2, 1) is $x + y - 3 = 0$. This line is a chord of another circle $C_2$ with centre (3, –2) and $AB = 4$. Find the radius of circle $C_2$.

A circle of unit radius touches positive $x$-axis and positive $y$-axis at $A$ and $B$ respectively. A variable line passing through origin intersects the circle in two points $D$ and $E$. If the slope of the line is $m$ and the area of triangle $DEB$ is maximum then $\dfrac{1}{m^2}$ is equal to:

Given equation of straight line x + 2y = 1 meets the coordinate axes at A and B. A circle is drawn through A, B and the origin. So, co-ordinates of points A (meet y-axis) and B (meet x-axis) are A(0, 1/2) and B(1, 0). Find the sum of distances of the point O(0, 0) from the tangent to the circle at origin and the tangent at B(1, 0), i.e., find l1 + l2.

If the lines $3x - 4y - 7 = 0$ and $2x - 3y - 5 = 0$ are two diameters of a circle of area $49\pi$ square units, the equation of the circle is

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:

A circle touches the line y = x at a point P such that OP = 4√2 where O is the origin. The circle contains the point (–10, 2) in its interior and the length of its chord on the line x + y = 0 is 6√2. Find the equation of the circle.

Two parallel chords of a circle S have length 10 and 14 and are 6 units apart. If a regular polygon of 12 sides is inscribed in a circle S, then find the area of regular polygon.

Let AB be a line segment of length 4 with $A$ on the line $y = 2x$ and $B$ on the line $y = x$. The locus of the middle point of the line segment is

The greatest distance of the point $P(10, 7)$ from the circle $x^2 + y^2 - 4x - 2y - 20 = 0$ is

From a point P outside a circle with centre at C, tangents PA and PB are drawn such that $\dfrac{1}{(CA)^2} + \dfrac{1}{(PA)^2} = \dfrac{1}{16}$, then the length of chord AB is

Find the values of $a$ for which the point $(2a, a+1)$ is an interior point of the larger segment of the circle $x^2 + y^2 - 2x - 2y - 8 = 0$ made by the chord whose equation is $x - y + 1 = 0$.

If $y + 3x = 0$ is the equation of a chord of the circle $x^2 + y^2 - 30x = 0$, then the equation of the circle with this chord as diameter is

If one of the diameters of the circle, given by the equation $x^2 + y^2 - 4x + 6y - 12 = 0$, is a chord of a circle S, whose centre is at $(-3, 2)$, then the radius of S is

Let $y=x+2$, $4y=3x+6$ and $3y=4x+1$ be three tangent lines to the circle $(x-h)^2+(y-k)^2=r^2$. Then $h+k$ is equal to:

A point P(x, y) is called a lattice point if x, y ∈ I (set of integers). Then the total number of lattice points in the interior of the circle $x^2 + y^2 = a^2$, $a \neq 0$ cannot be

A circle passes through $(-2, 4)$ and touches the $y$-axis at $(0, 2)$. Which one of the following equations can represent a diameter of this circle?

If a circle passing through the point $(-1, 0)$ touches $y$-axis at (0, 2), then the length of the chord of the circle along the $x$-axis is

In each of the following one or more options are correct. Choose the correct option(s).(e) The number of integers $(a \pm n, a \in \mathbb{N}$) lying inside the region bounded by the circles $x^2 + y^2 - 2x - 1 = 0$ and $x^2 + y^2 - 2x - 17 = 0$ is

A variable circle passes through the fixed point $A(p, q)$ and touches $x$-axis. The locus of the other end of the diameter through $A$ is

A circle is given by $x^2 + (y - 1)^2 = 1$, another circle C touches it externally and also the x-axis, then the locus of its centre is:

Coordinates of the centre of a circle, whose radius is 2 unit and which touches the line pair $x^2 - y^2 - 2x + 1 = 0$ are:

If the circle x2 + y2 + 4x + 22y + c = 0 bisects the circumference of the circle x2 + y2 - 2x + 8y - d = 0 (c, d > 0), then maximum value of cd is :

The middle point of the chord intercepted on the line lx + my + n = 0 by the circle x^2 + y^2 = a^2 is

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

Let the tangents drawn from the origin to the circle x^2 + y^2 - 8x - 4y + 16 = 0 touch it at the points A and B. The (AB)^2 is equal to

If a line, y = mx + c is a tangent to the circle, (x − 3)2 + y2 = 1 and it is perpendicular to a line L1, where L1 is the tangent to the circle, x2 + y2 = 1 at the point $\left(\frac{1}{2}, \frac{1}{2}\right)$, then find the relation between c.

The equation of normal to the circle $x^2 + y^2 - 5x + 2y - 48 = 0$ at the point $(5, 6)$ 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 common tangent to the circles $x^2 + y^2 = 4$ and $x^2 + y^2 + 6x + 8y - 24 = 0$ also passes through the point

The angle between the circles $S: x^2 + y^2 - 4x + 6y + 11 = 0$ and $S': x^2 + y^2 - 2x + 8y + 13 = 0$ is

The value of |a| for which the common chord of the circles $x^2 + y^2 = 8$ and $(x - a)^2 + y^2 = 8$ subtends a right angle at the origin are ............

If q be the angle between two tangents which are drawn to the circle $x^2 + y^2 - 6\sqrt{3}x - 6y + 27 = 0$ from the origin, then $2\sqrt{3}\tan q$ equals ............

A circle passing through the point $(3, \sqrt{2})$ touches the pair of lines $x^2 - y^2 - 2x + 1 = 0$. The centre of the circle is

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

If the circles $(x+1)^2+(y+2)^2=r^2$ and $x^2+y^2-4x-4y+4=0$ intersect at exactly two distinct points, then

The tangents at B(1, 7) and D(4, -2) to the circle $\omega$ : x2 + y2 - 2x - 4y - 20 = 0 meet at C. If the centre of the circle is A, then the area of $\square$ABCD (in sq. units) is

Let S = {(x, y) : (√5 - 1)x - √(10 + 2√5) y ≥ 0, (√5 - 1)x + √(10 + 12√5) y ≥ 0, x² + y² ≤ 9}then the diameter of S is:

Circles Questions (554)