Position of a Point W.R.T Ellipse

Q. A tangent to the ellipse $\frac{x^2}{25}+\frac{y^2}{16}=1$ at any point $P$ meets the line $x=0$ at a point $Q.$ Let $R$ be the image of $Q$ in the line $y=x$, then the circle whose extremities of a diameter are $Q$ and $R$ passes through a fixed point. The fixed point is

1. $(3,0)$
2. $(5,0)$
3. $(0,0)$
4. $(4,0)$

Q. A circle of radius $r(<a)$ is concentric with ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,(a>b)$, then slope of the common tangents to ellipse and circle is

1. $\pm \sqrt{\frac{r^2+b^2}{a^2+r^2}}$
2. $\pm \sqrt{\frac{r^2+b^2}{a^2-r^2}}$
3. $\pm \sqrt{\frac{r^2-b^2}{a^2+r^2}}$
4. $\pm \sqrt{\frac{r^2-b^2}{a^2-r^2}}$

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 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 on the ellipse
2. $S_1$$>0$$\Rightarrow (x_1,y_1)$ is inside 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. 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. 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. $\alpha \in (-\infty ,-\frac{5}{12})\cup (\frac{5}{12},\infty )$