Parametric Representation: Ellipse

Q. Let $P,Q,R$ be the points on the auxiliary circle of ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b)$ , such that $PQR$ is an equilateral triangle and $P^{\prime }{Q}^{\prime }{R}^{\prime }$ is corresponding triangle inscribed within the ellipse. Then centroid of the triangle $P^{\prime }{Q}^{\prime }{R}^{\prime }$ lies at

1. centre of the ellipse
2. focus of the ellipse
3. between focus and centre of the ellipse
4. between one extremity of minor axis and centre of the ellipse

Q. The equation of the curve whose parametric equations are $x=1+4\cos \theta ,y=2+3\sin \theta ,\theta \in R,$ is

1. $16x^2+9y^2-64x-18y-71=0$
2. $9x^2+16y^2-18x-64y-71=0$
3. $9x^2+16y^2-18x-64y+71=0$
4. $16x^2+9y^2-64x-18y+71=0$

Q. Let $P,D$ be two points on the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, whose eccentric angles differ by $\frac{\pi }{2}$. Then the locus of mid point of chord $PD$ is

1. $\frac{x^2}{a^2}+\frac{y^2}{b^2}=2$
2. $\frac{x^2}{a^2}+\frac{y^2}{b^2}=4$
3. $\frac{x^2}{a^2}+\frac{y^2}{b^2}=\frac{1}{4}$
4. $\frac{x^2}{a^2}+\frac{y^2}{b^2}=\frac{1}{2}$

Q. The ratio of the area enclosed by the locus of mid-point of PS and area of the ellipse where P is any point on the ellipse and S is the focus of the ellipse, is

1. $\frac{1}{2}$
2. $\frac{1}{3}$
3. $\frac{1}{5}$
4. $\frac{1}{4}$

Q. $P\left(\theta \right)$ and $Q(\theta +\frac{\pi }{2})$ are two points on the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1.$ The locus of midpoint of the chord PQ is

1. $\frac{x^2}{a^2}+\frac{y^2}{b^2}=\frac{1}{a}$
2. $\frac{x^2}{a^2}+\frac{y^2}{b^2}=\frac{1}{b}$
3. $\frac{x^2}{a^2}+\frac{y^2}{b^2}=\frac{1}{2}$
4. $\frac{x^2}{a^2}+\frac{y^2}{b^2}=\frac{1}{6}$

Q. Let $P,Q,R$ be the points on the auxiliary circle of ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b)$ , such that $PQR$ is an equilateral triangle and $P^{\prime }{Q}^{\prime }{R}^{\prime }$ is corresponding triangle inscribed within the ellipse. Then centroid of the triangle $P^{\prime }{Q}^{\prime }{R}^{\prime }$ lies at

1. centre of the ellipse
2. focus of the ellipse
3. between focus and centre of the ellipse
4. between one extremity of minor axis and centre of the ellipse

Q. If the chord joining the points $\alpha$ and $\beta$ on the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ subtends a right angle at (a, 0), then $\tan \left(\frac{\alpha }{2}\right)\tan \left(\frac{\beta }{2}\right)=$

1. $\frac{a^2}{b^2}$
2. $\frac{b^2}{a^2}$
3. $-\frac{a^2}{b^2}$
4. $-\frac{b^2}{a^2}$

Q. The area of the parallelogram formed by the tangents at the points whose eccentric angles are $\theta ,\theta +\frac{\pi }{2},\theta +\pi ,\theta +\frac{3\pi }{2}$ on the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ is

1. ab
2. 4ab
3. 3ab
4. 2ab