Parametric Form of Tangent: Ellipse

Q.

Tangents are drawn to the ellipse $x^2+2y^2=2$, then the locus of the mid-point of the intercept made by the tangents between the coordinate axis is

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

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

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

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

Q. If the tangent at point P on the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ meets the major axis at T, C is the centre and PN is the perpendicular on major axis, then the value of CN.CT is

1. $a^2$
2. $b^2$
3. $2a^2$
4. $2b^2$

Q.

If $CF$ is perpendicular from the centre $C$ of the ellipse $\frac{x^2}{49}+\frac{y^2}{25}=1$ on the tangent at any point $P$, and $G$ is the point where th e normal at $P$ meets the minor axis, then $(CF\cdot PG{)}^2$ is equal to___

Q. Let $P_i$ and $P_i^{\prime }$ be the feet of perpendiculars drawn from foci $S,S^{\prime }$ on a tangent $T_i$ to an ellipse whose length of semi major axis is $20,$ if $\sum _{i=1}^{10}\left(S{P}_i\right)\left(S{P}_i^{\prime }\right)=2560,$ then the value of eccentricity is

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

Q. If $p,p^{\prime }$ denote the lengths of the perpendiculars from the focus and the centre of an ellipse whose semi major axis is of length $a$ units on a tangent at a point on the ellipse and $r$ denotes the focal distance of the point, then

1. $rp=a{p}^{\prime }$
2. $rp+1=a{p}^{\prime }$
3. $ap=r{p}^{\prime }$
4. $ap=r{p}^{\prime }-1$

Q. If $\frac{x}{a}+\frac{y}{b}=\sqrt{2}$ touches the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,$ then its eccentric angle $\theta$ is equal to

1. $0^{\circ }$
2. ${90}^{\circ }$
3. ${45}^{\circ }$
4. ${60}^{\circ }$

Q. The tangent and normal to the ellipse $4x^2+9x^2=36$ at a point P on it meets the major axis in Q and R respectively. If QR = 4, then the eccentric angle of P is

1. ${\cos }^{-1}\frac{3}{5}$
2. ${\cos }^{-1}\frac{2}{3}$
3. ${\cos }^{-1}\frac{1}{3}$
4. ${\cos }^{-1}\frac{1}{5}$

Q. The value(s) of $\left[c\right]$ for which the line $y=4x+c$ touches the curve $x^2+16y^2=16$ is/are (where $[.]$ represents greatest integer function)

1. $-16$
2. $-17$
3. $16$
4. $17$

Q. If $p,p^{\prime }$ denote the lengths of the perpendiculars from the focus and the centre of an ellipse whose semi major axis is of length $a$ units on a tangent at a point on the ellipse and $r$ denotes the focal distance of the point, then

1. $rp=a{p}^{\prime }$
2. $rp+1=a{p}^{\prime }$
3. $ap=r{p}^{\prime }$
4. $ap=r{p}^{\prime }-1$

Q. If normal at $P(2,\frac{3\sqrt{3}}{2})$ meets the major axis of the ellipse $\frac{x^2}{16}+\frac{y^2}{9}=1$ at $Q$ and $S,S^{\prime }$ are foci of given ellipse along positive and negative directions of axes, then the ratio $SQ:S^{\prime }Q$ is

1. $\frac{7-\sqrt{8}}{7+\sqrt{8}}$
2. $\frac{8-\sqrt{7}}{8+\sqrt{7}}$
3. $\frac{7+\sqrt{8}}{7-\sqrt{8}}$
4. $\frac{8+\sqrt{7}}{8-\sqrt{7}}$

Q.

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

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

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

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

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

Q. $\mathrm{If}\mathrm{the}\mathrm{line}$ $3x+4y=\sqrt{7}\mathrm{touches}\mathrm{the}\mathrm{ellipse}3x^2+4y^2=1,$
$\mathrm{then}\mathrm{the}\mathrm{coordinates}\mathrm{of}\mathrm{the}\mathrm{point}\mathrm{of}\mathrm{contact}\mathrm{is}$ .

1. $(\frac{\sqrt{7}}{3},0)$
2. $(\frac{1}{\sqrt{7}},\frac{1}{\sqrt{7}})$
3. $\left(0,\frac{\sqrt{7}}{4}\right)$
4. undefined

Q. A tangent to $E_1:x^2+4y^2=4$ meets $E_2:x^2+2y^2=6$ at $P$ and $Q$. Tangents at $P$ and $Q$ of $E_2$ make an angle $\frac{\pi }{n}(n\in N)$. Then $n^2=$