find the equation of the circle which passes through the points 2,-2 and 3,4 and whose centre lies on the - Maths - Conic Sections

Question

find the equation of the circle which passes through the points
2,-2 and 3,4 and whose centre lies on the line x+y=2

ramandeep.singh · Accepted Answer

Equation of a circle with centre $(h, k)$ and radius r is given by:

$(x - h)^{2} + (y - k)^{2} = r^{2}$

Since the circle passes through (2,-2),

$∴ (2 - h)^{2} + (- 2 - k)^{2} = r^{2} ......... (1)$

Also, the circle passes through (3,4)

$∴ (3 - h)^{2} + (4 - k)^{2} = r^{2} ............ (2)$

Equating equations (1) and (2)

$(2 - h)^{2} + (- 2 - k)^{2} = (3 - h)^{2} + (4 - k)^{2} \Rightarrow 2 h + 12 k = 17......... (3)$

Also, it is given that the centre lies on the line $x + y = 2$ . Therefore, the centre (h,k) must satisfy the equation of the line.

$∴ h + k = 2.......... (4)$

From equations (3) and (4), we get

h = 0.7

k = 1.3

Putting these values in (2)

$(3 - 0.7)^{2} + (4 - 1.3)^{2} = r^{2} \Rightarrow r = 3.54 u n i t s$

Therefore, the equation of the circle is given by:

$(x - 0.7)^{2} + (y - 1.3)^{2} = {3.54}^{2}$

find the equation of the circle which passes through the points 2,-2 and 3,4 and whose centre lies on the line x+y=2