Question

If $PQR$ is an equilateral triangle inscribed in the auxiliary circle of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b)$ and $P^{\prime }{Q}^{\prime }{R}^{\prime }$ is the corresponding triangle inscribed within the ellipse then centroid of the triangle $P^{\prime }{Q}^{\prime }{R}^{\prime }$ lies at

• A. centre of ellipse
• B. focus of ellipse
• C. between focus and centre on major axis
• D. None of these

Answer 1

Answer: (A)

centre of ellipse

We have the ellipse :

$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,a>b$,

The auxiliary circle has equation: $x^2+y^2=a^2$

Parametric coordinate on auxiliary circle is:$(a\cos \alpha ,a\sin \alpha )$

Parametric coordinate of the same point when it is projected on the ellipse becomes:$(a\cos \alpha ,b\sin \alpha )$

According to figure:

We take one point P at an angle $\alpha$ from x axis.

Now as we know that $O$ is centre of $\Delta$, Points $QandR$ will be at a difference of $\pm 2\pi /3$ from $P$.

In Co-ordinate form:
P:$(a\cos \alpha ,a\sin \alpha )$

Q:$\left(acos\right(\alpha +\left(2\pi /3\right)),asin(\alpha +\left(2\pi /3\right)\left)\right)$

R:$\left(acos\right(\alpha -\left(2\pi /3\right)),asin(\alpha -\left(2\pi /3\right)\left)\right)$

Similarly,

$P^{\prime }$ :$\left(acos\right(\alpha ),bsin(\alpha \left)\right)$

$Q^{\prime }$: $\left(acos\right(\alpha +\left(2\pi /3\right)),bsin(\alpha +\left(2\pi /3\right)\left)\right)$

$R^{\prime }$ :$\left(acos\right(\alpha -\left(2\pi /3\right)),bsin(\alpha -\left(2\pi /3\right)\left)\right)$

Formula for centroid :$(\frac{x_1+x_2+x_3}{3},\frac{y_1+y_2+y_3}{3})$

Therefore putting values for $P^{\prime },Q^{\prime },R^{\prime }$, we get:

Centroid: $(0,0)$