Circle

Let $P$ be the point on the parabola $y = x^2$ such that the slope of the tangent to the parabola at the point $P$ is $4$. Let $Q$ be the point in the first quadrant lying on the circle $x^2 + y^2 = 2$ such that the slope of the tangent to ...

Let the straight line $y=2 x$ touch a circle with center $(0, \alpha), \alpha>0$, and radius $r$ at a point $A_1$. Let $B_1$ be the point on the circle such that the line segment $A_1 B_1$ is a diameter of the circle. Let $\alpha+r=5+\sqrt{...

Let $C_1$ be the circle of radius 1 with center at the origin. Let $C_2$ be the circle of radius $r$ with center at the point $A=(4,1)$, where $1 < r < 3$. Two distinct common tangents $P Q$ and $S T$ of $C_1$ and $C_2$ are drawn. The tange...

Let $A_1, A_2, A_3, \ldots, A_8$ be the vertices of a regular octagon that lie on a circle of radius 2 . Let $P$ be a point on the circle and let $P A_i$ denote the distance between the points $P$ and $A_i$ for $i=1,2, \ldots, 8$. If $P$ va...

Let $G$ be a circle of radius $R>0$. Let $G_{1}, G_{2}, \ldots, G_{n}$ be $n$ circles of equal radius $r>0$. Suppose each of the $n$ circles $G_{1}, G_{2}, \ldots, G_{n}$ touches the circle $G$ externally. Also, for $i=1,2, \ldots, n-1$, th...

Let \(A B C\) be the triangle with \(A B=1, A C=3\) and \(\angle B A C=\frac{\pi}{2}\). If a circle of radius \(r>0\) touches the sides \(A B, A C\) and also touches internally the circumcircle of the triangle \(A B C\), then the value of $...

Consider M with \(r = {{({2^{199}} - 1)\sqrt 2 } \over {{2^{198}}}}\). The number of all those circles Dn that are inside M is

Consider M with \(r = {{1025} \over {513}}\). Let k be the number of all those circles Cn that are inside M. Let l be the maximum possible number of circles among these k circles such that no two circles intersect. Then

Consider the region R = {(x, y) \(\in\) R \(\times\) R : x \(\ge\) 0 and y2 \(\le\) 4 \(-\) x}. Let F be the family of all circles that are contained in R and have centers on the x-axis. Let C be the circle that has largest radius among the...

Consider the region R = {(x, y) \(\in\) R \(\times\) R : x \(\ge\) 0 and y2 \(\le\) 4 \(-\) x}. Let F be the family of all circles that are contained in R and have centers on the x-axis. Let C be the circle that has largest radius among the...

Consider a triangle \(\Delta\) whose two sides lie on the x-axis and the line x + y + 1 = 0. If the orthocenter of \(\Delta\) is (1, 1), then the equation of the circle passing through the vertices of the triangle \(\Delta\) is

Let O be the centre of the circle x2 + y2 = r2, where \(r > {{\sqrt 5 } \over 2}\). Suppose PQ is a chord of this circle and the equation of the line passing through P and Q is 2x + 4y = 5. If the centre of the circumcircle of the triangle ...

A line y = mx + 1 intersects the circle \({(x - 3)^2} + {(y + 2)^2}\) = 25 at the points P and Q. If the midpoint of the line segment PQ has x-coordinate \(- {3 \over 5}\), then which one of the following options is correct?

Let the point B be the reflection of the point A(2, 3) with respect to the line \(8x - 6y - 23 = 0\). Let \(\Gamma_{A}\) and \(\Gamma_{B}\) be circles of radii 2 and 1 with centres A and B respectively. Let T be a common tangent to the ci...

Let S be the circle in the XY-plane defined the equation x2 + y2 = 4.Let E1E2 and F1F2 be the chords of S passing through the point P0 (1, 1) and parallel to the X-axis and the Y-axis, respectively. Let G1G2 be the chord of S passing throug...

Let RS be the diameter of the circle \({x^2}\, + \,{y^2} = 1\), where S is the point (1, 0). Let P be a variable point (other than R and S) on the circle and tangents to the circle at S and P meet at the point Q. The normal to the circle at...

A circle S passes through the point (0, 1) and is orthogonal to the circles \({(x - 1)^2}\, + \,{y^2} = 16\,\,and\,\,{x^2}\, + \,{y^2} = 1\). Then

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 tangent PT is drawn to the circle \({x^2}\, + {y^2} = 4\) at the point P \(\left( {\sqrt 3 ,1} \right)\). A straight line L, perpendicular to PT is a tangent to the circle \({(x - 3)^2}\) + \({y^2}\) = 1
A common tangent of the two circl...

A tangent PT is drawn to the circle \({x^2}\, + {y^2} = 4\) at the point P \(\left( {\sqrt 3 ,1} \right)\). A straight line L, perpendicular to PT is a tangent to the circle \({(x - 3)^2}\) + \({y^2}\) = 1.
A possible equation of L is

The locus of the mid-point of the chord of contact of tangents drawn from points lying on the straight line 4x - 5y = 20 to the circle \({x^2}\, + \,{y^2} = 9\) is

The straight line 2x - 3y = 1 divides the circular region \({x^2}\, + \,{y^2}\, \le \,6\) into two parts.
If $$S = \left\{ {\left( {2,\,{3 \over 4}} \right),\,\left( {{5 \over 2},\,{3 \over 4}} \right),\,\left( {{1 \over 4} - \,{1 \over 4}...

The circle passing through the point (-1, 0) and touching the y-axis at (0, 2) also passes through the point.

The centres of two circles \({C_1}\) and \({C_2}\) each of unit radius are at a distance of 6 units from each other. Let P be the mid point of the line segement joining the centres of \({C_1}\) and \({C_2}\) and C a circle touching circles ...

Tangents drawn from the point P (1, 8) to the circle
\({x^2}\, + \,{y^2}\, - \,6x\, - 4y\, - 11 = 0\)
touch the circle at the points A and B. The equation of the cirumcircle of the triangle PAB is

Consider
\(\,{L_1}:\,\,2x\,\, + \,\,3y\, + \,p\,\, - \,\,3 = 0\)
\(\,{L_2}:\,\,2x\,\, + \,\,3y\, + \,p\,\, + \,\,3 = 0\)
where p is a real number, and \(\,C:\,{x^2}\, + \,{y^2}\, + \,6x\, - 10y\, + \,30 = 0\)
STATEMENT-1 : If line $...

Equations of the sides QR, RP are

Points E and F are given by

The equation of circle C is

Match the statements in Column I with the properties Column II.

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Let \(\mathrm{ABCD}\) be a quadrilateral with area 18 , with side \(\mathrm{A B}\) parallel to the side \(\mathrm{C D}\) and \(\mathrm{A B}=2 \mathrm{CD}\). Let \(\mathrm{AD}\) be perpendicular to \(\mathrm{AB}\) and \(\mathrm{CD}\). If a c...

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

A circle touches the line $L$ and the circle $C_1$ externally such that both the circles are on the same side of the line, then the locus of center of the circle is:

A line $M$ through $A$ is drawn parallel to $B D$. Point $S$ moves such that its distances from
the line BD and the vertex A are equal. If locus of S cuts M at $\mathrm{T}_2$ and $\mathrm{T}_3$ and AC at $\mathrm{T}_1$, then area of $\Delta...

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

Circles with radii 3, 4 and 5 touch each other
externally if P is the point of intersection
of tangents to these circles at their points
of contact. Find the distance of P from the
point of contact.

Circles with radii 3, 4 and 5 touch each other externally. It P is the point of intersection of tangents to these circles at their points of contact, find the distance of P from the points of contact.

If one of the diameters of the circle \({x^2} + {y^2} - 2x - 6y + 6 = 0\) is a chord to the circle with centre (2, 1), then the radius of the circle is

Find the equation of circle touching the line 2x + 3y + 1 = 0 at (1, -1) and cutting orthogonally the circle having line segment joining (0, 3) and (- 2, -1) as diameter.

The centre of circle inscibed in square formed by the lines \({x^2} - 8x + 12 = 0\,\,and\,{y^2} - 14y + 45 = 0\), is

For the circle \({x^2}\, + \,{y^2} = {r^2}\), find the value of r for which the area enclosed by the tangents drawn from the point P (6, 8) to the circle and the chord of contact is maximum.

If \(a > 2b > 0\) then the positive value of \(m\) for which \(y = mx - b\sqrt {1 + {m^2}}\) is a common tangent to \({x^2} + {y^2} = {b^2}\) and \({\left( {x - a} \right)^2} + {y^2} = {b^2}\) is

If the tangent at the point P on the circle \({x^2} + {y^2} + 6x + 6y = 2\) meets a straight line 5x - 2y + 6 = 0 at a point Q on the y-axis, then the lenght of PQ is

Let A B be a chord of the circle \({x^2} + {y^2} = {r^2}\) subtending a right angle at the centre. Then the locus of the centriod of the triangle PAB as P moves on the circle is

Let PQ and RS be tangents at the extremities of the diameter PR of a circle of radius r. If PS and RQ intersect at a point X on the circumference of the circle, then 2r equals

Let \(\,2{x^2}\, + \,{y^2} - \,3xy = 0\) be the equation of a pair of tangents drawn from the origin O to a circle of radius 3 with centre in the first quadrant. If A is one of the points of contact, find the length of OA.

Let \(C_1\) and \(C_2\) be two circles with \(C_2\) lying inside \(C_1\). A circle C lying inside \(C_1\) touches \(C_1\) internally and \(C_2\) externally. Identify the locus of the centre of C.

If the circles \({x^2}\, + \,{y^2}\, + \,\,2x\, + \,2\,k\,y\,\, + \,6\,\, = \,\,0,\,\,{x^2}\, + \,\,{y^2}\, + \,2ky\, + \,k\, = \,0\) intersect orthogonally, then k is

The triangle PQR is inscribed in the circle \({x^2}\, + \,\,{y^2} = \,25\). If Q and R have co-ordinates (3, 4) and ( - 4, 3) respectively, then \(\angle \,Q\,P\,R\) is equal to

Let \({T_1}\), \({T_2}\) be two tangents drawn from (- 2, 0) onto the circle \(C:{x^2}\,\, + \,{y^2} = 1\). Determine the circles touching C and having \({T_1}\), \({T_2}\) as their pair of tangents. Further, find the equations of all possi...