1) Find the value of 'P', so that the points (2, 0), (0, 1), (4, 5) and (o, p) are concyclic.
Sol: Given Concyclic points are; (2, 0), (0, 1), (4, 5) and (0, p)
Let the circle be: x2 + y2 + 2gx + 2fy + c = 0
At: (2, 0): 4 + 0 + 2g (2) + 2f (0) + c = 0
⇒ 4g + c = -4 (1)
At (0, 1): 0 + 1 + 2g (0) + 2f (1) + c = 0
⇒ 2f + c = -1 (2)
At (4, 5): 16 + 25 + 2g (4) + 2f (5) + c = 0
⇒ 8g + 10f + c = - 41 (3)
2) Find the equation of the circle which passes through (4, 1), (6, 5) and whose centre lies on
4x + 3y - 24 = 0
Sol: Given points: (4, 1) and (6, 5)
Let the required circle be: x2 + y2 + 2gx + 2fy + c = 0
At (4, 1): 16 + 1 + 2g (4) + 2f (1) + c = 0
⇒ 8g + 2f + c = - 17 (1)
At (6, 5) : 36 + 25 + 2g (6) + 2f (5) + c = 0
⇒ 12g + 10f + c = - 61 (2)
Centre (-g, -f) lies on the line: 4x + 3y - 24 = 0
4 (-g) + 3 (-f) - 24 = 0
4g + 3f = - 24 (3)
3) Show that the circles x2 + y2 - 4x - 6y - 12 = 0 and x2 + y2 + 6x + 18y + 26 = 0 touch each other. Find the point of contact and the equation of the common tangent at the point of contact.
Sol: Given first circle; (S) = x2 + y2 - 4x - 6y - 12 = 0
Centre : C1 (2, 3).
Given second Circle; S' = x2 + y2 + 6x + 18y + 26 =0
Centre : C2 (-3, -9)
⇒ The given two circles touch each other externally
(4) Find the direct common tangents of x2 + y2 + 22x - 4y - 100 = 0 and
x2 + y2 - 22x + 4y + 100 = 0
Sol: Given first circle : (S) = x2 + y2 + 22x - 4y - 100 = 0
Centre : C1 (-11, 2)
Given second circle : (S') = x2 + y2 - 22x + 4y + 100 = 0
Centre : C2 (11, -2)
5) Find the equation of the circle with centre (-2, 3) cutting a chord length of 2 units on 3x + 4y + 4 = 0.
Sol: Given Centre : C (-2, 3)
6) Show that A (3, -1) lies on the circle x2 + y2 - 2x + 4y = 0 and find the other end of diameter through A.
7) Obtain the parametric equations of the circle x2 + y2 - 6x + 4y - 12 = 0
Sol: Given circle: x2 + y2 - 6x + 4y - 12 = 0
Centre : (3, -2)