Question

The value of $k$, if $(1,2),(k,1)$ are conjugate points with respect to the ellipse $2$$x^2$ +$3$$y^2$ = $6$

• A. $2$
• B. $0$
• C. $6$
• D. $8$

Answer 1

The correct option is B $0$

Given: Ellipse:$2x^2+3y^2=6$

$\Rightarrow \frac{x^2}{3}+\frac{y^2}{2}=1$----------1 $(where,a=\sqrt{3},b=\sqrt{2})$

Conjugate points:$(1,2)$ and $(k,1)$

Conjugate points: If polar one point passes through the other then the polar of other point will also pass through the first point.

Polar of $(1,2)$ is $\frac{x\left(1\right)}{3}+\frac{y\left(2\right)}{2}=1\Rightarrow \frac{x}{3}+y=1$---------2

Polar of $(k,1)$ is $\frac{x\left(k\right)}{3}+\frac{y\left(1\right)}{2}=1\Rightarrow \frac{kx}{3}+\frac{y}{2}=1$---------3

We know by definition of conjugate polar of point$(1,2)$ will lie on polar of $(k,1)$

So, Equation 2 & 3 $\Rightarrow$ For Equation 2, $\frac{k}{3}+1=1\Rightarrow k=0$