Question

A second-degree curve inscribing the $\Delta ABC,$ where the sides $AB,BC$ and $CA$ have the equations $L_1{L}_2+\lambda L_2{L}_3+\mu L_3{L}_1=0,$ where $\lambda$ and $\mu$ are parameters. Let the equations of $AB,BC$ and $CA$ be $y=2x,y+2x=0$, $x=1$, respectively.

The circumcentre of the $\Delta ABC$ is:

• A. $\left(\frac{5}{4},\frac{5}{8}\right)$
• B. $\left(\frac{5}{2},0\right)$
• C. $(0,\frac{5}{4})$
• D. $\left(\frac{5}{4},-\frac{5}{8}\right)$

Answer 1

Answer: (B)

$\left(\frac{5}{2},0\right)$

Given that $L_1:y=2x$
$L_2:y=-2x$
$L_3:x=1$
$L_1{L}_2+\lambda L_2{L}_3+\mu L_3{L}_1=0$
$\Rightarrow (y-2x)(y+2x)+\lambda (y+2x)(x-1)+\mu (y-2x)(x-1)=0$
$\Rightarrow (2\lambda -2\mu -4)x^2+y^2+(\lambda +\mu )xy+2(\mu -\lambda )x=0$
Since, above equation is circle with center $((\lambda -\mu ),0)$
Therefore, $2\lambda -2\mu -4=1$ and $\lambda +\mu =0$
$\Rightarrow \lambda -\mu =\frac{5}{2}$
Therefore, circumcentre is $(\frac{5}{2},0)$
Ans: B