Circles Questions (554)

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:

Two circles with centre at $A$ and $B$ touch at $T$. $BD$ is the tangent at $D$ and $TC$ is a common tangent. $AT$ has length 3 and $BT$ has length 2. The length of $CD$ is:

In a 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:

Considering the circles, $x^2 + y^2 = 25$ and $x^2 + y^2 = 9$. From the point $A(0, 5)$ two segments are drawn touching the inner circle at the points $B$ and $C$ while intersecting the outer circle at the points $D$ and $E$. If $O$ is the centre of both the circles then the length of the segment $OF$ that is perpendicular to $DE$, is:

Points $P$ and $Q$ are $3$ units apart. A circle centered at $P$ with a radius of $3$ units intersects a circle centered at $Q$ with radius $\sqrt{3}$ units at point $A$ and $B$. The area of the quadrilateral $APBQ$ is:

Circle $x^2 + y^2 + 16x + 12y + c = 0$ is touched by a straight line with slope $2$ and $y$-intercept $5$ units at a point $Q$. Then the coordinates of $Q$ are

Tangents are drawn at the point of intersections of the circles $x^2 + y^2 = 1$ and $x^2 + y^2 - (\lambda + 6)x + (8 - 2\lambda)y - 3 = 0$. ($\lambda$ being the variable). Then the locus of the point of intersection of these tangents 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 $DC = 12$ units then the length of $DE$ is:

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

The locus of the mid points of the chords of the circle $x^2 + y^2 - ax - by = 0$ which subtend a right angle at $(a/2, b/2)$ is:

If the two circles $C_1: x^2 + y^2 = 16$ and circle $C_2$ of radius $5$ units intersect in such a manner that the common chord of maximum length has a slope equal to $3/4$, then the coordinates of the centre of $C_2$ are:

The radius of circle $S_3$ is:

The area of circle circumscribing $x = \frac{-3h}{2}$ is:

Let $C_1, C_2, C_3$ be the centres of circles $S_1, S_2, S_3$ respectively, then which of the following must be true:

Let Q be a point from where tangents drawn to circle $g(x,y) = 0$ are mutually perpendicular. If A, B are the points of contact of tangent drawn from Q to circle $g(x,y) = 0$, then area of triangle QAB is:

The area of region bounded by circle $f(x,y) = 0$ with x-axis in the first quadrant is:

The number of points with positive integral coordinates satisfying $f(x,y) > 0, g(x,y) 3$ and $x < 6$ is:

The value of $x_2$ is:

The centre of circle $C_2$ is:

The $\lim_{n \to \infty} \frac{x_n}{2^n}$ is:

The locus of points of intersection of the tangents to $x^2 + y^2 = a^2$ at the extremities of a chord of circle $x^2 + y^2 = a^2$ which touches the circle $x^2 + y^2 - 2ax = 0$ is(are):

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

A circle touches the line $x + y - 2 = 0$ at $(1,1)$ and cuts the circle $x^2 + y^2 + 4x + 5y - 6 = 0$ at $P$ and $Q$. Then:

Let $x, y$ be real variable satisfy the $x^2 + y^2 + 8x - 10y - 40 = 0$. Let $a = \max\sqrt{(x+2)^2 + (y-3)^2}$ and $b = \min\sqrt{(x+2)^2 + (y-3)^2}$, then:

A square is inscribed in the circle $x^2 + y^2 - 2x + 4y - 93 = 0$ with the sides parallel to the coordinate axes. The coordinates of the vertices are:

Let $A, B, C, D$ lie on a line such that $AB = BC = CD = 1$. The points $A$ and $C$ are also joined by a semicircle with $AC$ as diameter and $P$ is a variable point on this semicircle such that $\angle PRD = 0, 0 \leq \pi \leq \pi$. Let $R$ is the region bounded by arc $AP$, the straight line $PD$ and line $AD$

Consider two circles $S_1$ and $S_2$ (externally touching) having centres at points $A$ and $B$ whose radii are 1 and 2 respectively. A tangent to circle $S_1$ from point $B$ touches the circle $S_1$ at point $C$. $D$ is chosen on circle $S_2$ so that $AC$ is parallel to $BD$ and two segments $BC$ and $AD$ do not intersect. Segment $AD$ intersect the circle $S_1$ at $E$. The line through $B$ and $E$ intersects the circle $S_1$ at another point $F$.

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

In the triangle $ABC$, the angle bisector $AK$ is perpendicular to the median $BM$ and $\angle ABC = 120°$, then:

There are two two circles in a parallelogram. One of them of radius 3units is inscribed in the parallelogram, and the other touches two sides of the parallelogram and the first circle. The distance between the points of tangency which lie on the same side of the parallelogram is equal to 3 units.

Point $M$ moved on the circle $(x-4)^2 + (y-8)^2 = 20$. Then it broke away from it and move along a tangent to the circle, cuts the $x$-axis at the point $(-2, 0)$. The coordinates of a point on the circle at which the moving point broke away is:

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):

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 $C$ be a circle with centre 'O' and $HK$ is the chord of contact of tangents drawn from a point $A$. $OA$ intersects the circle 'C' at $P$ and $Q$ and $B$ is the midpoint of $HK$, then:

Chords of the circle $x^2 + y^2 = 9$ are drawn such that segments intercepted from the chords by the curve $y^2 - 4x - 4y = 0$ subtend right angle at the origin. If the locus of the middle points of the chords with respect to circle is a curve $S$, then:

$A$ and $B$ are two points in $xy$ plane, which are $2\sqrt{2}$ unit distance apart and subtend an angle of $90°$ at $C(1,2)$ on the line $x - y + 1 = 0$ which is larger than any angle subtend by the line segment $AB$ at any other point on the line. The equation of the circle through the points $A$, $B$ and $C$ is:

A variable circle passes through the origin $O$ and cuts off portions $OP$ and $OQ$ from $x$-axis and $Y$-axis respectively such that $m(OP) + n(OQ)$ is equal to unity. If the circle passes through a fixed point $(x_1, y_1)$ other than $O$, then:

If two points $A(-2, a)$ and $B(4, \beta)$ are such that the triangle $AOB$ is the right-angle triangle right angle at $O$. If $S = 0$ be the equation of the locus of the foot of the perpendicular $P$ drawn from the point $O$ from the line $AB$, then:

Let $A, B, C$ and $D$ be four distinct point on a line in that order. The circles with diameter $AC$ is $x^2 + y^2 + ax + c = 0$ and $BD$ is $x^2 + y^2 - by = 0$ intersect at $X$ and $Y$ the line $XY$ meets $BC$ at $Z$. Let $P$ be a point on $XY$ other than $Z$, the line $CP$ intersects the circle with diameter $AC$ at $C$ and $M$, line $BP$ intersects the circle with diameter $BD$ at $B$ and $N$ and the equation of line $AM$ and $DN$ are $hx + cy + a = 0$ and $cx + ay + b = 0$ respectively, then which of the following is true (where $\omega$ is a cube root of unity)

An isosceles right angled triangle $ABC$ is such that $\angle B = 90°$, $AC = \sqrt{2}$ and $A$ and $C$ moves on positive coordinate axis, then

A point $M$ divides $A$ and $B$ in the ratio $1:2$ where $A$ and $B$ diametrically opposite ends of a circle $x^2 + y^2 - 5x - 9y + 22 = 0$ square $AMCD$ and $BMEF$ on the length $AM$ and $MB$ are constructed on the same side of line $AB$ if co-ordinates of $A$ is $(1, 3)$ then find the orthocentre of $\triangle ABE$.

If the circle passing through the distinct points $(a,t), (t,a)$ and $(r,t)$ for all values of '$r$' passes through a fixed point then

A line $L_1$ intersect $x$ and $y$ axes at $P$ and $Q$ respectively. Another line $L_2$, perpendicular to $L_1$, cuts $x$ and $y$ axes at $T$ and $S$ respectively. The locus of the point of intersection of the lines $PS$ and $QT$ is a circle passing through the

The equation of a circle of radius $1$ touching the circles $x^2 + y^2 - 2|x| = 0$ is:

A straight line through the vertex $P$ of a triangle $PQR$ intersect the side $QR$ at the point $S$ and the circumcircle of the triangle $PQR$ at the point $T$. If $S$ is not the centre of the circumcircle, then:

Let $L_1$ be a straight line passing through the origin and $L_2$ be the straight line $x + y = 1$. If the intercepts made by the circle $x^2 + y^2 - x + 3y = 0$ on $L_1$ and $L_2$ are equal, which of the following equations can represent $L_1$.

Two circles have centres at $(a,0)$ and $(-a,0)$ and radii $r_1$ and $r_2$ $(a > r_1 > r_2)$. Then the points of contact of the common tangents to two circles lies on the

A circle $S$ of radius unity touches a line $L$ at $P$. A point $A$ lies on $S$ and $N$ is the foot of the perpendicular from $A$ to $L$. The area of $\triangle PAN$ as $A$ varies cannot be equal 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:

Let $C_1$ and $C_2$ be centres of two circles whose radii are $2$ and $4$ respectively. Also $C_1C_2 = 10$ and direct common tangents of these circles touch them at $P, Q, R, S$. Another circle of radius $\lambda$ is drawn passing through $P, Q, R, S$. Then