Question

Let φ(x, y) = 0 be the equation of a circle. If φ(0, λ)=0 has equal roots$\lambda = \frac { 4 } { 5 } , 5$then the centre of the circle is

• A. $\left( 2 , \frac { 29 } { 10 } \right)$
• B. $\left( \frac { 29 } { 10 } , 2 \right)$
• C. $\left( - 2 , \frac { 29 } { 10 } \right)$
• D. None of these

Answer 1

Answer: (B)

$\left( \frac { 29 } { 10 } , 2 \right)$

Answer 2

Let φ $( x , y ) \equiv x ^ { 2 } + y ^ { 2 } + 2 g x + 2 f y + c = 0$

$\therefore \phi ( 0 , \lambda ) = 0 + \lambda ^ { 2 } + 0 + 2 \mathrm { f } \lambda + \mathrm { c } = 0$ have equal roots.

then

$\therefore \mathrm { f } = - 2 \& \mathrm { c } = 4$ &

$\therefore \lambda ^ { 2 } + 2 \mathrm {~g} \lambda + \mathrm { c } = 0$

Here

$\therefore$ $\lambda ^ { 2 } + 2 g \lambda + 4 = 0$ have roots $\frac { 4 } { 5 } , 5$

$\therefore \frac { 4 } { 5 } + 5 = - 2 \mathrm {~g}$

⇒ $\mathrm { g } = - \frac { 29 } { 10 }$

$\therefore$Centre = $( - \mathrm { g } , - \mathrm { f } ) = \left( \frac { 29 } { 10 } , 2 \right)$