Position of a Point W.R.T Ellipse

Q.

Find the set of values of $\alpha$ for which point the $P(\alpha ,-\alpha )$ is inside
$\frac{x^2}{16}+\frac{y^2}{9}=1$

1. $(\frac{-12}{5},\frac{12}{5})$
2. $(\frac{-13}{5},\frac{13}{5})$
3. $[\frac{-12}{5},\frac{12}{5}]$
4. $(\frac{-13}{5},\frac{13}{5}]$

Q.

If for the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,S_1=\frac{x_1^2}{a^2}+\frac{y_1^2}{b^2}-1$, for point $(x_1,y_1)$, Which of the following is true

1. $S_1$$>0$$\Rightarrow (x_1,y_1)$ is inside the ellipse
2. $S_1=0\Rightarrow (x_1,y_1)$ is on the ellipse
3. $S_1$$<0$$\Rightarrow (x_1,y_1)$is outside the ellipse
4. $S_1$$>0$$\Rightarrow (x_1,y_1)$ is outside the ellipse

Q. The point (4, -3) lies inside the ellipse $5x^2+7y^2=140$.

1. False
2. True

Q. The number of tangents that can be drawn to the ellipse $16x^2+9y^2=144$ from the point (3, -4) is .

1. 2
2. 3
3. \N
4. 1

Q. An ellipse with major and minor axis length as $2a$ and $2b$ units touches coordinate axis in first quadrant. If foci are $(x_1,y_1)$ and $(x_2,y_2),$ then the value of $x_1{x}_2+y_1{y}_2$ is

1. $2b^2$
2. $a^2{b}^2$
3. $a^2+b^2$
4. $2a^2$

Q. If the point $P(\alpha ,-\alpha )$ lies inside the ellipse $\frac{x^2}{16}+\frac{y^2}{9}=1,$ then

1. $\alpha \in (-\infty ,-\frac{12}{5})\cup (\frac{12}{5},\infty )$
2. $\alpha \in (-\frac{12}{5},\frac{12}{5})$
3. $\alpha \in (-\frac{5}{12},\frac{5}{12})$
4. $\alpha \in (-\infty ,-\frac{5}{12})\cup (\frac{5}{12},\infty )$

Q. The position of the point $(2,-3)$ with respect to the ellipse $\frac{x^2}{9}+\frac{y^2}{25}=1,$ is

1. It lies on the ellipse
2. It lies outside the ellipse
3. It lies inside the ellipse
4. None of these

Q. If the tangent at the point $P\left(\theta \right)$ to the ellipse $16x^2+11y^2=256$ is also a tangent to the circle $x^2+y^2-2x=15,$ then possible value(s) of $\theta$ is/are

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

Q. The line $3x+4y=\sqrt{7}$ touches the ellipse $3x^2+4y^2=1$ at the point

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

Q. If tangents drawn to the ellipse at the parametric point $\theta$, where $\tan \theta =2$ meets the auxillary circle at $P$ and $Q$ and $PQ$ subtends rightangle at the centre of the ellipse, then eccentricity of the ellipse is

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