Parametric Form of Normal : Ellipse

Q.

Let a tangent be drawn to the ellipse $\left(\frac{x^2}{27}\right)+y^2=1$ at $(3\sqrt{3}\cos \theta ,\sin \theta )$ where $\theta \in \left(0,\frac{\pi }{2}\right)$. Then the value of θ such that the sum of intercepts on axes made by a tangent is minimum is equal to:

1. $\frac{\pi }{8}$

2. $\frac{\pi }{6}$

3. $\frac{\pi }{3}$

4. $\frac{\pi }{4}$

Q. Let the line $y=mx$ and the ellipse $2x^2+y^2=1$ intersect at point $P$ in the first quadrant. If the normal to this ellipse at $P$ meets the co-ordinate axes at $(-\frac{1}{3\sqrt{2}},0)$ and $(0,\beta )$, then $\beta$ is equal to:

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

Q.

Find an equation equivalent to $x^2-y^2=4$ in polar coordinates.

Q. Tangents are drawn to the ellipse $\frac{x^2}{16}+\frac{y^2}{7}=1$ at the end points of the latus rectum. The area of quadrilateral formed by these tangents is

1. $\frac{64}{3}$ sq. units
2. $\frac{128}{3}$ sq. units
3. $\frac{32}{3}$ sq. units
4. $\frac{256}{3}$ sq. units

Q. The area of the rectangle (in sq. units) formed by the perpendiculars drawn from the centre of the ellipse having major axis and minor axis lengths as $2a$ and $2b$ units respectively to the tangent and normal at a point whose eccentric angle is $\frac{\pi }{4}$ is

1. $\frac{(a^2-b^2)ab}{a^2+b^2}$
2. $\frac{(a^2+b^2)ab}{a^2-b^2}$
3. $\frac{(a^2-b^2)}{ab(a^2+b^2)}$
4. $\frac{(a^2+b^2)}{(a^2-b^2)ab}$

Q. If $CF$ be the perpendicular from the centre $C$ of the ellipse $\frac{x^2}{12}+\frac{y^2}{8}=1,$ on the tangent at any point $P$ and $G$ is the point where the normal at $P$ meets the major axis, then the value of $CF\cdot PG=$

Q.

The normal at a point P on the ellipse $x^2+4y^2=16$ meets the X - axis at Q. If M is the mid-point of the line segment PQ, then the locus of M intersects the latusrectum of the given ellipse at the points

1. $(\pm \frac{3\sqrt{5}}{2},\pm \frac{2}{7})$

2. $(\pm \frac{3\sqrt{5}}{2},\pm \frac{\sqrt{19}}{4})$

3. $(\pm 2\sqrt{3},\pm \frac{1}{7})$

4. $(\pm 2\sqrt{3},\pm \frac{4\sqrt{3}}{7})$

Q. The set of real values of $x$ for which
${\log }_{0.2}\left(\frac{x+2}{x}\right)\le 1$ is

Q. If the normal at $\theta$ on the ellipse $5x^2+14y^2=70$ cuts the curve again at a point $2\theta$, then $\cos \theta$ =

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

Q. If $\beta$ is one of the angles between the normals to the ellipse, $x^2+3y^2=9$ at the points $(3\cos \theta ,\sqrt{3}\sin \theta )$ and $(-3\sin \theta ,\sqrt{3}\cos \theta )$; $\theta \in (0,\frac{\pi }{2});$ then $\frac{2\cot \beta }{\sin 2\theta }$ is equal to

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

Q.

The set of real values of $x$ for which expression
${\log }_{12}(x^2-x)<1$ holds true is

Q. Tangents are drawn to the ellipse $\frac{x^2}{16}+\frac{y^2}{7}=1$ at the end points of the latus rectum. The area of quadrilateral formed by these tangents is

1. $\frac{64}{3}$ sq. units
2. $\frac{128}{3}$ sq. units
3. $\frac{256}{3}$ sq. units
4. $\frac{32}{3}$ sq. units

Q. Find the point on the ellipse $\frac{x^2}{4}+\frac{y^2}{36}=1$ whose eccentric angle is $\frac{5\pi }{4}$ radian.

1. $P(\sqrt{2},-3\sqrt{2})$
2. $P(\sqrt{2},3\sqrt{2})$
3. $P(-\sqrt{2},3\sqrt{2})$
4. $P(-\sqrt{2},-3\sqrt{2})$

Q. If $\sqrt{{\log }_{2}x}-0.5={\log }_2\sqrt{x}$, then $x$ equals to

Q. The normal at $P\left(\theta \right)$ and D$(\theta +\frac{\pi }{2})$ meet the major axis of $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ at Q and R. Then $P{Q}^2+D{R}^2=$

1. $b^2(1-e^2)(2-e{)}^2$
2. $a^2(1-e^2)(2-e{)}^2$
3. $a^2(1+e^2)(2+e^2)$
4. $b^2(1+e^2)(2+e^2)$

Q. The eccentricity of an ellipse whose centre is at the origin is $\frac{1}{2}.$ If one of its directrices is $x=-4,$ then the equation of the normal to it at $(1,\frac{3}{2})$ is:

1. $2y-x=2$
2. $4x-2y=1$
3. $4x+2y=7$
4. $x+2y=4$