Circle

If two distinct chords, drawn from the point (p, q) on the circle \({x^2}\, + \,{y^2} = \,px\, + \,qy\,\,(\,where\,pq\, \ne \,0)\) are bisected by the x - axis, then

\(C_1\) and \(C_2\) are two concentric circles, the radius of \(C_2\) being twice that of \(C_1\). From a point P on \(C_2\), tangents PA and PB are drawn to \(C_1\). Prove that the centroid of the triangle PAB lies on \(C_1\).

The number of common tangents to the circles \({x^2}\, + \,{y^2} = 4\) and \({x^2}\, + \,{y^2}\, - 6x\, - 8y = 24\) is

If the circle \({x^2}\, + \,{y^2} = \,{a^2}\) intersects the hyperbola \(xy = {c^2}\) in four points \(P\,({x_1},\,{y_1}),\,Q\,\,({x_2},\,{y_2}),\,\,R\,({x_3},\,{y_3}),\,S\,({x_4},\,{y_4}),\) then

For each natural number k, let \({C_k}\) denote the circle with radius k centimetres and centre at the origin. On the circle \({C_k}\), a-particle moves k centimetres in the counter-clockwise direction. After completing its motion on $${C_k...

The chords of contact of the pair of tangents drawn from each point on the line 2x + y = 4 to circle \({x^2} + {y^2} = 1\) pass through the point........................

Let C be any circle with centre \(\,\left( {0\, , \sqrt {2} } \right)\). Prove that at the most two rational points can to there on C. (A rational point is a point both of whose coordinates are rational numbers.)

The angle between a pair of tangents drawn from a point P to the circle \({x^2}\, + \,{y^2}\, + \,\,4x\, - \,6\,y\, + \,9\,{\sin ^2}\,\alpha \, + \,13\,{\cos ^2}\,\alpha \, = \,0\) is \(2\,\alpha\).
The equation of the locus of the point...

The intercept on the line y = x by the circle \({x^2} + {y^2} - 2x = 0\) is AB. Equation of the circle with AB as a diameter is................................

A circle passes through three points A, B and C with the line segment AC as its diameter. A line passing through A angles DAB and CAB are \(\,\alpha \,\,and\,\,\beta\) respectively and the distance between the point A and the mid point of ...

Find the intervals of value of a for which the line y + x = 0 bisects two chords drawn from a point \(\left( {{{1\, + \,\sqrt 2 a} \over 2},\,{{1\, - \,\sqrt 2 a} \over 2}} \right)\) to the circle $$\,\,2{x^2}\, + \,2{y^2} - (\,1\, + \sqrt ...

The circles \({x^2} + {y^2} - 10x + 16 = 0\) and \({x^2} + {y^2} = {r^2}\) intersect each other in two distinct points if

The equation of the locus of the mid-points of the circle \(4{x^2} + 4{y^2} - 12x + 4y + 1 = 0\) that subtend an angle of \(2\pi /3\) at its centre is.................................

Consider a family of circles passing through two fixed points A (3, 7) and B (6, 5). Show that the chords on which the circle \({x^2}\, + \,{y^2} - \,4x - \,6y - 3 = 0\) cuts the members of the family are concurrent at a point. Find the coo...

The locus of the centre of a circle, which touches externally the circle \({x^2} + {y^2} - 6x - 6y + 14 = 0\) and also touches the y-axis, is given by the equation:

Find the coordinates of the point at which the circles \({x^2}\, + \,{y^2} - \,4x - \,2y = - 4\,\,and\,\,{x^2}\, + \,{y^2} - \,12x - \,8y = - 36\) touch each other. Also find equations common tangests touching the circles in the distinct ...

Let a circle be given by 2x (x - a) + y (2y - b) = 0, \((a\, \ne \,0,\,\,b\, \ne 0)\). Find the condition on a abd b if two chords, each bisected by the x-axis, can be drawn to the circle from \(\left( {a,\,\,{b \over 2}} \right)\).

The centre of a circle passing through the points (0, 0), (1, 0) and touching the circle \({x^2} + {y^2} = 9\)is

If a circle passes through the points of intersection of the coordinate axes with the lines \(\lambda \,x - y + 1 = 0\) and x - 2y + 3 = 0, then the value of \(\lambda\) = .........

Two circles, each of radius 5 units, touch each other at (1, 2). If the equation of their common tangent is 4x + 3y = 10, find the equation of the circles.

A circle touches the line y = x at a point P such that OP = \({4\sqrt 2 \,}\), where O is the origin. The circle contains the point (- 10, 2) in its interior and the length of its chord on the line x + y = 0 is \({6\sqrt 2 \,}\). Determine ...

The line x + 3y = 0 is a diameter of the circle \({x^2} + {y^2} - 6x + 2y = 0\,\).

The area of the triangle formed by the positive x-axis and the normal and the tangent to the circle \({x^2} + {y^2} = 4\,\,at\,\,\left( {1,\sqrt 3 } \right)\) is,..................

If \(\left( {{m_i},{1 \over {{m_i}}}} \right),\,{m_i}\, > \,0,\,i\, = 1,\,2,\,3,\,4\) are four distinct points on a circle, then show that \({m_1}\,{m_2}\,{m_3}\,{m_4}\, = 1\)

The lines 2x - 3y = 5 and 3x - 4y = 7 are diameters of a circle of area 154 sq. units. Then the equation of this circle is

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

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

If the circle \({C_1}:{x^2} + {y^2} = 16\) intersects another circle \({C_2}\) of radius 5 in such a manner that common chord is of maximum lenght and has a slope equal to 3/4, then the coordinates of the centre of \({C_2}\) are...............

The equations of the tangents drawn from the origin to the circle \({x^2}\, + \,{y^2}\, - \,2rx\,\, - 2hy\, + {h^2} = 0\), are

Let a given line \(L_1\) intersects the x and y axes at P and Q, respectively. Let another line \(L_2\), perpendicular to \(L_1\), cut the x and y axes at R and S, respectively. Show that the locus of the point of intersection of the lines ...

The circle \({x^2}\, + \,{y^2} - \,4x\, - 4y + \,4 = 0\) is inscribed in a triangle which has two of its sides along the co-ordinate axes. The locus of the circumcentre of the triangle is $$x\, + \,y\, - xy\, + k\,{\left( {{x^2}\, + \,{y^2}...

The area of the triangle formed by the tangents from the point (4, 3) to the circle \({x^2} + {y^2} = 9\) and the line joining their points of contact is...................

The equation of the line passing through the points of intersection of the circles \(3{x^2} + 3{y^2} - 2x + 12y - 9 = 0\) and \({x^2} + {y^2} - 6x + 2y - 15 = 0\) is..............................

Lines 5x + 12y - 10 = 0 and 5x - 12y - 40 = 0 touch a circle \(C_1\) of diameter 6. If the centre of \(C_1\) lies in the first quadrant, find the equation of the circle \(C_2\) which is concentric with \(C_1\) and cuts intercepts of length ...

From the point A(0, 3) on the circle \({x^2} + 4x + {(y - 3)^2} = 0\), a chord AB is drawn and extended to a point M such that AM = 2AB. The equation of the locus of M is..........................

No tangent can be drawn from the point (5/2, 1) to the circumcircle of the triangle with vertices \(\left( {1,\sqrt 3 } \right)\,\,\left( {1, - \sqrt 3 } \right),\,\,\left( {3,\sqrt 3 } \right)\).

Let \({x^2} + {y^2} - 4x - 2y - 11 = 0\) be a circle. A pair of tangentas from the point (4, 5) with a pair of radi from a quadrilateral of area............................

From the origin chords are drawn to the circle \({(x - 1)^2} + {y^2} = 1\). The equation of the locus of the mid-points of these chords is.............

The locus of the mid-point of a chord of the circle \({x^2} + {y^2} = 4\) which subtends a right angle at the origin is

The abscissa of the two points A and B are the roots of the equation \({x^2}\, + \,2ax\, - {b^2} = 0\) and their ordinates are the roots of the equation \({x^2}\, + \,2px\, - {q^2} = 0\). Find the equation and the radius of the circle with ...

The lines 3x - 4y + 4 = 0 and 6x - 8y - 7 = 0 are tangents to the same circle. The radius of this circle is ........................................

Through a fixed point (h, k) secants are drawn to the circle \(\,{x^2}\, + \,{y^2} = \,{r^2}\). Show that the locus of the mid-points of the secants intercepted by the circle is \(\,{x^2}\, + \,{y^2}\) = \(hx + ky\).

The centre of the circle passing through the point (0, 1) and touching the curve \(\,y = {x^2}\) at (2, 4) is

The equation of the circle passing through (1, 1) and the points of intersection of \({x^2} + {y^2} + 13x - 3y = 0\) and \(2{x^2} + 2{y^2} + 4x - 7y - 25 = 0\) is

The point of intersection of the line 4x - 3y - 10 = 0 and the circle \({x^2} + {y^2} - 2x + 4y - 20 = 0\) are ........................and ...................

If A and B are points in the plane such that PA/PB = k (constant) for all P on a given circle, then the value of k cannot be equal to ..........................................

Find the equations of the circle passing through (- 4, 3) and touching the lines x + y = 2 and x - y = 2.

Let A be the centre of the circle \({x^2}\, + \,{y^2}\, - \,2x\,\, - 4y\, - 20 = 0\,\). Suppose that the tangents at the points B (1, 7) and D (4. - 2) on the circle meet at the point C. Find the area of the quadrilateral ABCD.

A square is inscribed in the circle \({x^2} + {y^2} - 2x + 4y + 3 = 0\). Its sides are parallel to the coordinate axes. The one vertex of the square is

Two circles \({x^2} + {y^2} = 6\) and \({x^2} + {y^2} - 6x + 8 = 0\) are given. Then the equation of the circle through their points of intersection and the point (1, 1) is