Circles Questions (554)

Let the line $L:\sqrt{2}x+y=\alpha$ pass through the point of intersection $P$ (in the first quadrant) of the circle $x^2+y^2=3$ and the parabola $x^2=2y$. Let $L$ touch two circles $C_1$ and $C_2$ of equal radius $2\sqrt{3}$. If the centres $Q_1$ and $Q_2$ of $C_1$ and $C_2$ lie on the $y$-axis, then the square of the area of the triangle $PQ_1Q_2$ is equal to

Given a line segment AB, where 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

Two Circles of radii 36 units and 9 units touch each other externally, a third circle of radius $r$ touches the two given circles externally and also their common tangent, then the value of $r$ is:

The circle x2 + y2 = 4x + 8y = 5 intersects the line 3x - 4y = m at two distinct points. Find the range of m.

Three concentric circles of which the biggest is $x^2 + y^2 = 1$, have their radii in A.P. If the line $y = x + 1$ cuts all the circles in real and distinct points. The interval in which the common difference of the A.P. will be:

The equation of the smallest circle passing through the points of intersection of the line \(x + y = 1\) and the circle \(x^2 + y^2 = 9\) 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:

The value of a for which the point (a, a + 2) is an interior point of smaller segment of the curve \(x^2 + y^2 - 4 = 0\) made by the chord of the curve whose equation is \(3x + 4y + 12 = 0\)

Statement-1: Tangents are drawn from the point \((17, 7)\) to the circle \(x^2 + y^2 = 169\). The tangents are mutually perpendicular.becauseStatement-2: The locus of the points from which mutually perpendicular tangents can be drawn to the given circle is \(x^2 + y^2 = 338\).

Let $C$ be the circle of minimum area touching the parabola $y=6-x^2$ and the lines $y=\sqrt{3}|x|$. Then which one of the following points lies on the circle $C$?

Suppose two perpendicular tangents can be drawn from the origin to the circle x2 + y2 - 6x - 2cy + 17 = 0, for some real c, then |c| is equal to :

Let a circle C pass through the points (4, 2) and (0, 2), and its centre lie on 3x + 2y + 2 = 0. Then the length of the chord, of the circle C , whose mid-point is (1, 2), is :

In the given figure $AB$ is tangent at $A$ to the circle with centre at $O$; point $D$ is interior to circle and $DB$ intersects the circle at $C$. If $BC = DC = 3$, $OD = 2$ and $AB = 6$, then find the value of $[r]$ (where $r$ is the radius of circle and $[.]$ represent G.I.F)

Let $C$ be a circle with radius $\sqrt{10}$ units and centre at the origin. Let the line $x+y=2$ intersect the circle $C$ at the points $P$ and $Q$. Let $MN$ be a chord of $C$ of length 2 units and slope $-1$. Then a distance (in units) between the chord $PQ$ and the chord $MN$ is:

Let circle C be the image of x + y - 2x + 4y - 4 = 0 in the line 2x - 3y + 5 = 0 and A be the point on C 2 2 such that OA is parallel to x-axis and A lies on the right hand side of the centre O of C . If B(\alpha, \beta), with \beta < 4, lies on C such that the length of the are AB is (1/6) th of the perimeter of C , then \beta - \sqrt3\alpha is equal to

The number of common tangents, to the circles $x^2+y^2-18x-15y+131=0$ and $x^2+y^2-6x-6y-7=0$, is

Equations of two diameters of a circle are $2x-3y=5$ and $3x-4y=7$. The line joining the points $\left(-\frac{22}{7},-4\right)$ and $\left(-\frac{1}{7},3\right)$ intersects the circle at only one point $P(\alpha,\beta)$. Then $17\beta-\alpha$ is equal to

Let the equation of the circle, which touches $x$-axis at the point $(a, 0)$, $a > 0$ and cuts off an intercept of length $b$ on $y$-axis be $x^2 + y^2 - \alpha x - \beta y + \gamma = 0$. If the circle lies below $x$-axis, then the ordered pair $(2a, b^2)$ is equal to

If a line having y-intercept \('c'\) makes a chord of length \('a'\) to the circle \(x^2 + y^2 = a^2\), then:

The equation(s) of the tangent at the point (0, 0) to the circle, making intercepts of lengths 2a and 2b units on the coordinate axes, is/are:

Two vertices of a triangle are (0, 2) and (4, 3). If its orthocentre is at the origin, then its third vertex lies in which quadrant?

If the two circles \((x-1)^2 + (y-3)^2 = r^2\) and \(x^2 + y^2 - 8x + 2y + 8 = 0\) intersect in two distinct points, then

Four distinct points $(2k,3k)$, $(1,0)$, $(0,1)$ and $(0,0)$ lie on a circle for $k$ equal to:

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 \(\frac{a}{4}

22. Through a random point \((p, q)\) on the cartesian plane secants are drawn to the circle \(x^2 + y^2 = r^2\). If the locus of mid-point of the secants to the circle is \(x^2 + 2hxy + y^2 + 2gx + 2fy + c = 0\). Then:

Let $ABCD$ and $AEFG$ be squares of side 4 and 2 units, respectively. The point $E$ is on the line segment $AB$ and the point $F$ is on the diagonal $AC$. Then the radius $r$ of the circle passing through the point $F$ and touching the line segments $BC$ and $CD$ satisfies:

Let △ABC be inscribed in a circle having radius unity. The three internal bisectors of the angles A, B and C are extended to intersect the circumcircle of △ABC at A₁, B₁ and C₁ respectively. Then (AA₁ cos(A/2) + BB₁ cos(B/2) + CC₁ cos(C/2))/(sin A + sin B + sin C) =

Consider a family of circles which are passing through the point (−1, 1) and are tangent to x-axis. If (h, k) are the co-ordinates of the centre of the circles, then the set of values of k is given by the interval:

Let the set of all values of $r$, for which the circles $(x+1)^2+(y+4)^2=r^2$ and $x^2+y^2-4x-2y-4=0$ intersect at two distinct points be the interval $(\alpha,\beta)$. Then $\alpha\beta$ is equal to

Locus of the point of intersection of the tangents at the ends of the normal chords of the parabola y2 = 4ax is :

If the circles x2 + y2 - 16x - 20y + 164 = r2 and \((x-4) ^2+(y-7)^{2}=36\) intersect at two distinct points, then

Let a circle passing through $(2,0)$ have its centre at the point $(h,k)$. Let $(x_c,y_c)$ be the point of intersection of the lines $3x+5y=1$ and $(2+c)x+5c^2y=1$. If $h=\lim_{c\to1}x_c$ and $k=\lim_{c\to1}y_c$, then the equation of the circle is:

Let a variable line passing through the centre of the circle $x^2+y^2-16x-4y=0$ meet the positive coordinate axes at the points $A$ and $B$. Then the minimum value of $OA+OB$, where $O$ is the origin, is equal to

Let circle $C$ be the image of $x^2 + y^2 - 2x + 4y - 4 = 0$ in the line $2x - 3y + 5 = 0$ and $A$ be the point on $C$ such that $OA$ is parallel to $x$-axis and $A$ lies on the right hand side of the centre $O$ of $C$. If $B(\alpha, \beta)$, with $\beta < 4$, lies on $C$ such that the length of the arc $AB$ is $\tfrac{1}{6}^{\text{th}}$ of the perimeter of $C$, then $\beta - \sqrt{3}\,\alpha$ is equal to

Let the line x + y = 1 meet the circle x 2 + y 2 = 4 at the points A and B . If the line perpendicular to AB and passing through the mid point of the chord AB intersects the circle at C and D , then the area of the quadrilateral ADBC is equal to :

Two circles in the first quadrant of radii $r_1$ and $r_2$ touch the coordinate axes. Each of them cuts off an intercept of $2$ units with the line $x+y=2$. Then $r_1^2+r_2^2-r_1 r_2$ is equal to ____.

Let PQ and MN be two straight lines touching the circle $x^2+y^2-4x-6y-3=0$ at the points A and B respectively. Let O be the centre of the circle and $\angle AOB=\pi/3$. Then the locus of the point of intersection of the lines PQ and MN is:

If a variable line \(3 x+4 y-\lambda=0\) is such that the two circles \(x^{2}+y^{2}-2 x-2 y+1=0\) and \(x^{2}+y^{2}-18 x-2 y+78=0\) are on its opposite sides, then the set of all values of \(\lambda\) is the interval

The chord nearest to the centre of the circle will be the longest chord. From options, the given lines are at distances $\frac{2}{3}, \frac{3}{5}, \frac{2}{3}, \frac{7}{5}$ respectively from the centre $(3,4)$.

Consider a circle $C_1:\ x^2+y^2-4x-2y=\alpha-5$. Let its mirror image in the line $y=2x+1$ be another circle $C_2:\ 5x^2+5y^2-10fx-10gy+36=0$. Let $r$ be the radius of $C_2$. Then $\alpha+r$ is equal to ________.

From the following figure, which depicts the given situation, the circle touches the line \(y = -x\). Also, from the graph, the radius is obtained as \(4 - k\). The circle passes through \((0, 4)\) and its centre is at \((0, k)\). The radius of the circle is:

If two chords of the circle \(x^2 + y^2 - ax - by = 0\), drawn from the point \((a, b)\) is divided by the x-axis in the ratio 2:1 then:

The radius of the circle passing through the vertices of the triangle ABC, is

Given a line segment AB, where A is at (0, 0) and B at (a, 0). Three circles S₁, S₂, S₃ of radius R are centred at the endpoints and the midpoint of the line segment AB. A fourth circle S₄ is drawn touching the 3 given circles.If \(0

If a point \(P\) has coordinates \((0, -2)\) and \(Q\) is any point on the circle, \(x^2 + y^2 - 5x - y + 5 = 0\), then the maximum value of \((PQ)^2\) is

If the incentre of an equilateral triangle is (1, 1) and the equation of its one side is \(3x + 4y + 3 = 0\), then the equation of the circumcircle of this triangle is

The circle \(x^2 + y^2 = 1\) cuts the x-axis at P and Q. Another circle with centre at Q and variable radius intersects to first circle at R above the x-axis and the line segment PQ at S. The maximum area of the triangle QSR is

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

The length of the diameter of the circle which touches the x-axis at the point (1, 0) and passes through the point (2, 3) is

Three sides of a triangle have the equations \(L_i \equiv y - m_i x = 0\); \(i = 1, 2\). Then \(L_1 L_2 + \lambda L_2 L_3 + \mu L_3 L_1 = 0\), where \(\lambda \neq 0, \mu \neq 0\), is the equation of the circumcircle of the triangle if