Point Form of Normal: Ellipse

Q.

The sum of the distance of a point $\left(2,-3\right)$ from the foci of an ellipse $16{\left(x-2\right)}^2+25{\left(y+3\right)}^2=400$ is

1. $8$

2. $6$

3. $50$

4. $32$

5. $10$

Q. The normal at a point $P$ on the ellipse $x^2+4y^2=16$ intersects the $x$-axis at $Q$. If $M$ is the midpoint of the line segment $PQ$, then the locus of $M$ intersects the latus rectum of the given ellipse at 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. If normal at point $(6,2)$ to the ellipse passes through its nearest focus $(5,2),$ having centre at $(4,2),$ then its eccentricity is

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

Q. How many tangents to the circle $x^2+y^2=3$ are there which are normal to the ellipse $\frac{x^2}{9}+\frac{y^2}{4}=1$

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

Q. The eccentricity of an ellipse with center 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

Q. The equation of normal at point $P(8\sqrt{2},1)$ on the ellipse $\frac{x^2}{144}+\frac{y^2}{9}=1$ is

1. $\sqrt{2}x+y=15$
2. $\sqrt{2}x-y=15$
3. $x+\sqrt{2}y=15$
4. $x-\sqrt{2}y=15$

Q. If the normal at an end point of a latus rectum of an ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b)$ passes through one extremity of the minor axis. Then the eccentricity of the ellipse is equal to

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

Q. The equation of the normal to the ellipse $\frac{x^2}{18}+\frac{y^2}{8}=1$ at the point (3, 2) is .

1. 3x - 2y = 5
2. 3x + 2y = 5
3. 2x - 3y = 5
4. 2x + 3y = 5