Circle

Let C₁ and C₂ be the centres of the circles x² + y² – 2x – 2y – 2 = 0 and x² + y² – 6x – 6y + 14 = 0 respectively. If P and Q are the points of intersection of these circles, then the area (in sq. units) of the quadrilateral PC₁QC₂ is :

The centres of those circles which touch the circle, $${x^2} + {y^2} - 8x - 8y - 4 = 0$$, externally and also touch the $$x$$-axis, lie on :

The equation of a circle with origin as a center and passing through an equilateral triangle whose median is of length $$3$$$$a$$ is :

The point diametrically opposite to the point $$P(1, 0)$$ on the circle $${x^2} + {y^2} + 2x + 4y - 3 = 0$$ is :

Three distinct points A, B and C are given in the 2 -dimensional coordinates plane such that the ratio of the distance of any one of them from the point $$(1, 0)$$ to the distance from the point $$(-1, 0)$$ is equal to $${1 \over 3}$$. Then the circumcentre of the triangle ABC is at the point :

Locus of the image of the point $$(2, 3)$$ in the line $$\left( {2x - 3y + 4} \right) + k\left( {x - 2y + 3} \right) = 0,\,k \in R,$$ is a :

The two adjacent sides of a cyclic quadrilateral are 2 and 5 and the angle between them is 60^o. If the area of the quadrilateral is $$4\sqrt 3$$, then the perimeter of the quadrilateral is :

A rectangle is inscribed in a circle with a diameter lying along the line 3y = x + 7. If the two adjacent vertices of the rectangle are (–8, 5) and (6, 5), then the area of the rectangle (in sq. units) is :

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 :

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| such that $$A \cup B \subseteq C$$ is equal to

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 relation from A $$\cap$$ B to A $$\cap$$ C is 2^p, then the value of p is :

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 :

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 :

Let a circle passing through $$(2,0)$$ have its centre at the point $$(\mathrm{h}, \mathrm{k})$$. Let $$(x_{\mathrm{c}}, y_{\mathrm{c}})$$ be the point of intersection of the lines $$3 x+5 y=1$$ and $$(2+\mathrm{c}) x+5 \mathrm{c}^2 y=1$$. If \mathrm{h}=\lim _\limits{\mathrm{c} \rightarrow 1} x_{\mathrm{c}} and \mathrm{k}=\lim _\limits{\mathrm{c} \rightarrow 1} y_{\mathrm{c}}, then the equation of the circle is :

A square is inscribed in the circle $$x^2+y^2-10 x-6 y+30=0$$. One side of this square is parallel to $$y=x+3$$. If $$\left(x_i, y_i\right)$$ are the vertices of the square, then $$\Sigma\left(x_i^2+y_i^2\right)$$ is equal to:

If the image of the point $$(-4,5)$$ in the line $$x+2 y=2$$ lies on the circle $$(x+4)^2+(y-3)^2=r^2$$, then $$r$$ is equal to:

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 :

A circle is inscribed in an equilateral triangle of side of length 12. If the area and perimeter of any square inscribed in this circle are $$m$$ and $$n$$, respectively, then $$m+n^2$$ is equal to

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, is equal to :