Circle

Let a circle \(C_{1}\) be obtained on rolling the circle \(x^{2}+y^{2}-4 x-6 y+11=0\) upwards 4 units on the tangent \(\mathrm{T}\) to it at the point \((3,2)\). Let \(C_{2}\) be the image of \(C_{1}\) in \(\mathrm{T}\). Let \(A\) and \(B\)...

The set of all values of $a^{2}$ for which the line $x+y=0$ bisects two distinct chords drawn from a point $\mathrm{P}\left(\frac{1+a}{2}, \frac{1-a}{2}\right)$ on the circle $2 x^{2}+2 y^{2}-(1+a) x-(1-a) y=0$, is equal to :

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 :

Let $P\left(a_1, b_1\right)$ and $Q\left(a_2, b_2\right)$ be two distinct points on a circle with center $C(\sqrt{2}, \sqrt{3})$. Let $\mathrm{O}$ be the origin and $\mathrm{OC}$ be perpendicular to both $\mathrm{CP}$ and $\mathrm{CQ}$. If ...

Let the tangents at the points \(A(4,-11)\) and \(B(8,-5)\) on the circle \(x^{2}+y^{2}-3 x+10 y-15=0\), intersect at the point \(C\). Then the radius of the circle, whose centre is \(C\) and the line joining \(A\) and \(B\) is its tangent,...

A circle with centre (2, 3) and radius 4 intersects the line \(x+y=3\) at the points P and Q. If the tangents at P and Q intersect at the point \(S(\alpha,\beta)\), then \(4\alpha-7\beta\) is equal to ___________.

The points of intersection of the line \(ax + by = 0,(a \ne b)\) and the circle \({x^2} + {y^2} - 2x = 0\) are \(A(\alpha ,0)\) and \(B(1,\beta )\). The image of the circle with AB as a diameter in the line \(x + y + 2 = 0\) is :

Points P(\(-\)3, 2), Q(9, 10) and R(\(\alpha,4\)) lie on a circle C and PR as its diameter. The tangents to C at the points Q and R intersect at the point S. If S lies on the line \(2x-ky=1\), then k is equal to ____________.

The locus of the mid points of the chords of the circle \({C_1}:{(x - 4)^2} + {(y - 5)^2} = 4\) which subtend an angle \({\theta _i}\) at the centre of the circle \(C_1\), is a circle of radius \(r_i\). If $${\theta _1} = {\pi \over 3},{\t...

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

Let the centre of a circle C be \((\alpha, \beta)\) and its radius \(r < 8\). Let \(3 x+4 y=24\) and \(3 x-4 y=32\) be two tangents and \(4 x+3 y=1\) be a normal to C. Then \((\alpha-\beta+r)\) 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 ___________.

A line segment AB of length \(\lambda\) moves such that the points A and B remain on the periphery of a circle of radius \(\lambda\). Then the locus of the point, that divides the line segment AB in the ratio 2 : 3, is a circle of radius :

Let A be the point \((1,2)\) and B be any point on the curve \(x^{2}+y^{2}=16\). If the centre of the locus of the point P, which divides the line segment \(\mathrm{AB}\) in the ratio \(3: 2\) is the point C\((\alpha, \beta)\), then the len...

Consider three circles:
\({C_1}:{x^2} + {y^2} = {r^2}\)
\({C_2}:{(x - 1)^2} + {(y - 1)^2} = {r^2}\)
\({C_3}:{(x - 2)^2} + {(y - 1)^2} = {r^2}\)
If a line L : y = mx + c be a common tangent to C1, C2 and C3 such that C1 and C3 lie on one sid...

Let the tangent to the circle C1 : x2 + y2 = 2 at the point M(\(-\)1, 1) intersect the circle C2 : (x \(-\) 3)2 + (y \(-\) 2)2 = 5, at two distinct points A and B. If the tangents to C2 at the points A and B intersect at N, then the area of...

Let a triangle ABC be inscribed in the circle \({x^2} - \sqrt 2 (x + y) + {y^2} = 0\) such that \(\angle BAC = {\pi \over 2}\). If the length of side AB is \(\sqrt 2\), then the area of the \(\Delta\)ABC is equal to :

Let the mirror image of a circle \(c_{1}: x^{2}+y^{2}-2 x-6 y+\alpha=0\) in line \(y=x+1\) be \(c_{2}: 5 x^{2}+5 y^{2}+10 g x+10 f y+38=0\). If \(\mathrm{r}\) is the radius of circle \(\mathrm{c}_{2}\), then \(\alpha+6 \mathrm{r}^{2}\) is e...

\(\text { Let } S=\left\{(x, y) \in \mathbb{N} \times \mathbb{N}: 9(x-3)^{2}+16(y-4)^{2} \leq 144\right\}\) and
\(T=\left\{(x, y) \in \mathbb{R} \times \mathbb{R}:(x-7)^{2}+(y-4)^{2} \leq 36\right\}\). Then \(n(S \cap T)\) is equal to _____...

Let \(A B\) be a chord of length 12 of the circle \((x-2)^{2}+(y+1)^{2}=\frac{169}{4}\). If tangents drawn to the circle at points \(A\) and \(B\) intersect at the point \(P\), then five times the distance of point \(P\) from chord \(A B\) ...

Let the lines \(y + 2x = \sqrt {11} + 7\sqrt 7\) and \(2y + x = 2\sqrt {11} + 6\sqrt 7\) be normal to a circle \(C:{(x - h)^2} + {(y - k)^2} = {r^2}\). If the line \(\sqrt {11} y - 3x = {{5\sqrt {77} } \over 3} + 11\) is tangent to the ...

If the tangents drawn at the points \(O(0,0)\) and \(P\left( {1 + \sqrt 5 ,2} \right)\) on the circle \({x^2} + {y^2} - 2x - 4y = 0\) intersect at the point Q, then the area of the triangle OPQ is equal to :

If one of the diameters of the circle \({x^2} + {y^2} - 2\sqrt 2 x - 6\sqrt 2 y + 14 = 0\) is a chord of the circle \({(x - 2\sqrt 2 )^2} + {(y - 2\sqrt 2 )^2} = {r^2}\), then the value of r2 is equal to ____________.

Let \(C\) be the centre of the circle \(x^{2}+y^{2}-x+2 y=\frac{11}{4}\) and \(P\) be a point on the circle. A line passes through the point \(\mathrm{C}\), makes an angle of \(\frac{\pi}{4}\) with the line \(\mathrm{CP}\) and intersects th...

For \(\mathrm{t} \in(0,2 \pi)\), if \(\mathrm{ABC}\) is an equilateral triangle with vertices \(\mathrm{A}(\sin t,-\cos \mathrm{t}), \mathrm{B}(\operatorname{cost}, \sin t)\) and \(C(a, b)\) such that its orthocentre lies on a circle with c...

Let the tangents at two points \(\mathrm{A}\) and \(\mathrm{B}\) on the circle \(x^{2}+\mathrm{y}^{2}-4 x+3=0\) meet at origin \(\mathrm{O}(0,0)\). Then the area of the triangle \(\mathrm{OAB}\) is :

A rectangle R with end points of one of its sides as (1, 2) and (3, 6) is inscribed in a circle. If the equation of a diameter of the circle is 2x \(-\) y + 4 = 0, then the area of R is ____________.

Let a circle C of radius 5 lie below the x-axis. The line L1 : 4x + 3y + 2 = 0 passes through the centre P of the circle C and intersects the line L2 = 3x \(-\) 4y \(-\) 11 = 0 at Q. The line L2 touches C at the point Q. Then the distance o...

The set of values of k, for which the circle \(C:4{x^2} + 4{y^2} - 12x + 8y + k = 0\) lies inside the fourth quadrant and the point \(\left( {1, - {1 \over 3}} \right)\) lies on or inside the circle C, is :

If the circle \(x^{2}+y^{2}-2 g x+6 y-19 c=0, g, c \in \mathbb{R}\) passes through the point \((6,1)\) and its centre lies on the line \(x-2 c y=8\), then the length of intercept made by the circle on \(x\)-axis is :

A circle \(C_{1}\) passes through the origin \(\mathrm{O}\) and has diameter 4 on the positive \(x\)-axis. The line \(y=2 x\) gives a chord \(\mathrm{OA}\) of circle \(\mathrm{C}_{1}\). Let \(\mathrm{C}_{2}\) be the circle with $$\mathrm{OA...

Let C be a circle passing through the points A(2, \(-\)1) and B(3, 4). The line segment AB s not a diameter of C. If r is the radius of C and its centre lies on the circle \({(x - 5)^2} + {(y - 1)^2} = {{13} \over 2}\), then r2 is equal to ...

Let the abscissae of the two points \(P\) and \(Q\) on a circle be the roots of \(x^{2}-4 x-6=0\) and the ordinates of \(\mathrm{P}\) and \(\mathrm{Q}\) be the roots of \(y^{2}+2 y-7=0\). If \(\mathrm{PQ}\) is a diameter of the circle $$x^{...

Let the abscissae of the two points P and Q be the roots of \(2{x^2} - rx + p = 0\) and the ordinates of P and Q be the roots of \({x^2} - sx - q = 0\). If the equation of the circle described on PQ as diameter is $$2({x^2} + {y^2}) - 11x -...

Let a circle C touch the lines \({L_1}:4x - 3y + {K_1} = 0\) and \({L_2} = 4x - 3y + {K_2} = 0\), \({K_1},{K_2} \in R\). If a line passing through the centre of the circle C intersects L1 at \(( - 1,2)\) and L2 at \((3, - 6)\), then the equ...

A circle touches both the y-axis and the line x + y = 0. Then the locus of its center is :

If the circles \({x^2} + {y^2} + 6x + 8y + 16 = 0\) and \({x^2} + {y^2} + 2\left( {3 - \sqrt 3 } \right)x + 2\left( {4 - \sqrt 6 } \right)y = k + 6\sqrt 3 + 8\sqrt 6\), \(k > 0\), touch internally at the point \(P(\alpha ,\beta )\), then ...

Let a circle C : (x \(-\) h)2 + (y \(-\) k)2 = r2, k > 0, touch the x-axis at (1, 0). If the line x + y = 0 intersects the circle C at P and Q such that the length of the chord PQ is 2, then the value of h + k + r is equal to ___________.

If the variable line 3x + 4y = \(\alpha\) lies between the two circles (x \(-\) 1)2 + (y \(-\) 1)2 = 1 and (x \(-\) 9)2 + (y \(-\) 1)2 = 4, without intercepting a chord on either circle, then the sum of all the integral values of \(\alpha\)...

Let B be the centre of the circle x2 + y2 \(-\) 2x + 4y + 1 = 0. Let the tangents at two points P and Q on the circle intersect at the point A(3, 1). Then 8.\(\left( {{{area\,\Delta APQ} \over {area\,\Delta BPQ}}} \right)\) is equal to ____...

Let \(A = \{ (x,y) \in R \times R|2{x^2} + 2{y^2} - 2x - 2y = 1\}\), \(B = \{ (x,y) \in R \times R|4{x^2} + 4{y^2} - 16y + 7 = 0\}\) and \(C = \{ (x,y) \in R \times R|{x^2} + {y^2} - 4x - 2y + 5 \le {r^2}\}\).Then the minimum value of |r...

Let P and Q be two distinct points on a circle which has center at C(2, 3) and which passes through origin O. If OC is perpendicular to both the line segments CP and CQ, then the set {P, Q} is equal to :

Two tangents are drawn from the point P(\(-\)1, 1) to the circle x2 + y2 \(-\) 2x \(-\) 6y + 6 = 0. If these tangents touch the circle at points A and B, and if D is a point on the circle such that length of the segments AB and AD are equal...

Consider a circle C which touches the y-axis at (0, 6) and cuts off an intercept \(6\sqrt 5\) on the x-axis. Then the radius of the circle C is equal to :

Let the equation x2 + y2 + px + (1 \(-\) p)y + 5 = 0 represent circles of varying radius r \(\in\) (0, 5]. Then the number of elements in the set S = {q : q = p2 and q is an integer} is __________.

Two circles each of radius 5 units touch each other at the point (1, 2). If the equation of their common tangent is 4x + 3y = 10, and C1(\(\alpha\), \(\beta\)) and C2(\(\gamma\), \(\delta\)), C1 \(\ne\) C2 are their centres, then |($$\alpha...

Let Z be the set of all integers,\(A = \{ (x,y) \in Z \times Z:{(x - 2)^2} + {y^2} \le 4\}\)\(B = \{ (x,y) \in Z \times Z:{x^2} + {y^2} \le 4\}\)\(C = \{ (x,y) \in Z \times Z:{(x - 2)^2} + {(y - 2)^2} \le 4\}\)If the total number of rela...

In the circle given below, let OA = 1 unit, OB = 13 unit and PQ \(\bot\) OB. Then, the area of the triangle PQB (in square units) is :

Let A(1, 4) and B(1, \(-\)5) be two points. Let P be a point on the circle (x \(-\) 1)2 + (y \(-\) 1)2 = 1 such that (PA)2 + (PB)2 have maximum value, then the points, P, A and B lie on :

If the locus of the mid-point of the line segment from the point (3, 2) to a point on the circle, x2 + y2 = 1 is a circle of radius r, then r is equal to :