The circle x2 + y2 – 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(x2 + y2)1/2 = 0. Find k.
The given circle is x2 + y2 – 4x – 4y + 4 = 0. This can be re-written as (x– 2)2 + (y – 2)2 = 4 which has centre C (2, 2) and radius 2.
Let the equation of third side is
Length of perpendicular from (2, 2) on AB = radius = CM
Since origin and (2, 2) lie on the same side of AB
Hence AB is the diameter of the circle passing through âˆ†OAB, mid point of AB is the centre of the circle i.e.,
Substituting the values of a and b in (1) then
∴ Locus of M (h, k) is
x + y – xy +
Hence the required value of k is 1.
Find the condition on a, b, c such that two chords of the circle
x2 + y2 – 2ax – 2by + a2 + b2 – c2 = 0
passing through the point (a, b + c) are bisected by the line y = x.
Find the limiting points of the circles
(x2 + y2 + 2gx + c) + λ(x2 + y2 + 2fy + d) = 0
The circle x2 + y2 – 4x – 8y + 16 = 0 rolls up the tangent to it at by 2 units, assuming the x-axis as horizontal, find the equation of the circle in the new position.
Find the equation of the circle of minimum radius which contains the three circles
x2 – y2 – 4y – 5 = 0
x2 + y2 + 12x + 4y + 31 = 0
and x2 + y2 + 6x + 12y + 36 = 0
Find the equation of the circle passing through (1, 0) and (0, 1) and having the smallest possible radius.
If the line x cos α + y sin α = p cuts the circle x2 + y2 = a2 in M and N, then show that the circle, whose diameter is MN, is
A line meets the co-ordinate axes at A and B. A circle is circumscribed about the triangle OAB. If the distance of the points A and B from the tangent at O, the origin, to the circle are m and n respectively, find the equation of the circle.
Tangents PQ, PR are drawn to the circle x2 + y2 = 36 from the point P(–8, 2) touching the circle at Q, R respectively. Find the equation of the circumcircle of the âˆ†PQR.
Let C1 and C2 be two circles with C2 lying inside C1. A circle C lying inside C1 touches C1 internally and C2 externally. Identify the locus of the centre of C.
Tangents TP and TQ are drawn from a point T to the circle x2 + y2 = a2. If the point T lies on the line px + qy = r, find the locus of centre of the circumcircle of triangle TPQ.