Let ABCD be a quadrilateral with area 18, with side AB parallel to CD and AB = 2CD. Let AD be perpendicular to AB and CD. If a circle is drawn inside the quadrilateral ABCD touching all the sides, then its radius is
Let A (0, 0), B (2a, 0), C (a, 2r), D (0, 2r). Since the circle touches the parallel sides AB and CD its diameter is 2r so that the radius of the circle is r. Since it touches AD, distance of the centre from AB and AD is r and the coordinates of the centre are (r, r).
or 2rx + ay – 4ar = 0
As the circle touches BC, distance of the centre from BC is also r.
From (1) and (2) we get a = 3, r = 2
So the required radius is 2.
Equation of the circle which passes through the origin, has its centre on the line x + y = 4 and cuts the circle
x2 + y2 – 4x + 2y + 4 = 0 orthogonally, is
If O is the origin and OP, OQ are distinct tangents to the circle x2 + y2 + 2gx + 2fy + c = 0, the circumcentre of the triangle OPQ is
The circle passing through the distinct points (1, t), (t, 1) and (t, t) for all values of t, passes through the point
If OA and OB are the tangents from the origin to the circle x2 + y2 + 2gx + 2fy + c = 0, and C is the centre of the circle, the area of the quadrilateral OACB is
The angle between a pair of tangents drawn from a point P to the circle x2 + y2 + 4x – 6y + 9 sin2α + 13 cos2α = 0 is 2α. The equation of the locus of the point P is
Equation of a circle through the origin and belonging to the co-axial system, of which the limiting points are (1, 2), (4, 3) is
If a line segment AM = a moves in the plane XOY remaining parallel toOX so that the left end point A slides along the circle x2 + y2 = a2, the locus of M is
If common chord of the circle C with centre at (2, 1) and radius r and the circle x2 + y2 – 2x – 6y + 6 = 0 is a diameter of the second circle, then the value of r is
Tangents drawn from the point P(1, 8) to the circle x2 + y2 – 6x – 4y – 11 = 0 touch the circle at the points A and B. The equation of the circumcircle of the triangle in PAB is
An equilateral triangle is inscribed in the circle x2 + y2 = a2 with the vertex at (a, 0). The equation of the side opposite to this vertex is