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

Let the equation of the
required circle be (*x* – *h*)^{2} + (*y*
– *k*)^{2} = *r*^{2}.

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

(4 – *h*)^{2}
+ (1 – *k*)^{2} = *r*^{2} …
(1)

(6 – *h*)^{2}
+ (5 – *k*)^{2} = *r*^{2} …
(2)

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

4*h* + *k* =
16 … (3)

From equations (1) and (2), we obtain

(4 – *h*)^{2}
+ (1 – *k*)^{2 }= (6 – *h*)^{2}
+ (5 – *k*)^{2}

⇒ 16 – 8*h*
+ *h*^{2} + 1 – 2*k* + *k*^{2} =
36 – 12*h* + *h*^{2} + 25 – 10*k*
+ *k*^{2}

⇒ 16 – 8*h*
+ 1 – 2*k* = 36 – 12*h* + 25 – 10*k*

⇒ 4*h* + 8*k*
= 44

⇒ *h* + 2*k*
= 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} = *r*^{2}

⇒ (1)^{2}
+ (– 3)^{2} = *r*^{2}

⇒ 1 + 9 = *r*^{2
}

⇒ *r*^{2}
= 10

⇒

Thus, the equation of the required circle is

(*x* – 3)^{2}
+ (*y* – 4)^{2} =

*x*^{2} –
6*x* + 9 + *y*^{2} – 8*y* + 16 = 10

*x*^{2} +
*y*^{2} – 6*x* – 8*y* + 15 = 0

**
**