Question

The equation of the circle whose diameter lies on $2 x + 3 y = 3$and $16 x - y = 4$ which passes through (4,6) is.

• A. $5 \left( x ^ { 2 } + y ^ { 2 } \right) - 3 x - 8 y = 200$
• B. $x ^ { 2 } + y ^ { 2 } - 4 x - 8 y = 200$
• C. $5 \left( x ^ { 2 } + y ^ { 2 } \right) - 4 x = 200$
• D. $x ^ { 2 } + y ^ { 2 } = 40$

Answer 1

Answer: (A)

$5 \left( x ^ { 2 } + y ^ { 2 } \right) - 3 x - 8 y = 200$

Answer 2

Let point $\left( x _ { 1 } , y _ { 1 } \right)$ on the diameter.

$\Rightarrow 2 x _ { 1 } + 3 y _ { 1 } = 3$ ….(i)

$16 x _ { 1 } - y _ { 1 } = 4$ ….(ii)

On solving (i) and (ii), we get centre,

$\Rightarrow x _ { 1 } = \frac { 3 } { 10 } , y _ { 1 } = \frac { 4 } { 5 }$

$\therefore$ Equation of circle,

$\left( x - x _ { 1 } \right) ^ { 2 } + \left( y - y _ { 1 } \right) ^ { 2 } = r ^ { 2 } \Rightarrow \left( x - \frac { 3 } { 10 } \right) ^ { 2 } + \left( y - \frac { 4 } { 5 } \right) ^ { 2 } = r ^ { 2 }$

$\bullet \bullet$ Circle passes through (4, 6).

So, $r ^ { 2 } = \left( \frac { 37 } { 10 } \right) ^ { 2 } + \left( \frac { 26 } { 5 } \right) ^ { 2 } \Rightarrow r ^ { 2 } = \frac { 4073 } { 100 }$

$\therefore$ Required equation of circle is

$\left( x - \frac { 3 } { 10 } \right) ^ { 2 } + \left( y - \frac { 4 } { 5 } \right) ^ { 2 } = \frac { 4073 } { 100 }$

$\Rightarrow 5 \left( x ^ { 2 } + y ^ { 2 } \right) - 3 x - 8 y = 200$.