# Find the equation of the circle passing through the points (4,1) and (6,5) and whose centre is on the line 4x + y = 16.

Let the equation of the required circle be (x – h)2+ (y – k)2 =r2

Since, the circle passes through points (4,1) and (6,5),

(4 – h)2+ (1 – k)2 =r2 .................(1)

(6– h)2+ (5 – k)2 =r2 ..................(2)

Since, the centre (h,k) of the circle lies on line 4x+y = 16,

4h + k =16..................... (3)

From the equation (1) and (2), we obtain

(4 – h)2+ (1 – k)2 =(6 – h)2 + (5 – k)2 16 – 8h + h2 +1 -2k +k2 = 36 -12h +h2+15 – 10k + k2 16 – 8h +1 -2k + 12h -25 -10k 4h +8k = 44 h + 2k =11................ (4)

On solving equations (3) and (4), we obtain h=3 and k= 4.

On substituting the values of h and k in equation (1), we obtain

(4 – 3)2+ (1 – 4)2 =r2 (1)2 + (-3)2 = r2 1+9 = r2 r = Thus, the equation of the requires circle is

(x – 3)2+ (y – 4)2 = x2 -6x + 9 + y2 – 8y + 16 =10

⇒ x2 + y2 -6x – 8y = 15 =0

Rate this question :

How useful is this solution?
We strive to provide quality solutions. Please rate us to serve you better.
Related Videos  Focal chord of parabola29 mins  Quiz on properties of focal chord of parabola36 mins  Interactive Quiz on Equation of Parabola41 mins  Lecture on Equation of Parabola59 mins
Try our Mini CourseMaster Important Topics in 7 DaysLearn from IITians, NITians, Doctors & Academic Experts
Dedicated counsellor for each student
24X7 Doubt Resolution
Daily Report Card
Detailed Performance Evaluation view all courses 