Point Form of Normal: Ellipse

Q. If the normal at the end of latus rectum of an ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ of eccentricity e passes through one end of the minor axis then $e^4+e^2=$

1. 
2. 
3. 
4. 

Q. C is the centre of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ and L is an end of a latus rectum. If the normal at L meets the major axis in G, then CG =

1. 
2. 
3. 
4. 

Q.

Find the equation of a normal to the ellipse $\frac{x^2}{16}+\frac{y^2}{9}=2$ at the point (4, 3).

1. 4x - 3y = 7

2. 4x + 3y = 25

3. 3x - 4y = 0

4. 3x + 4y = 24

Q.

What is the equation of normal to the ellipse
$\frac{x^2}{25}+\frac{y^2}{16}=2\text{at}(5,4).$

1. 5x - 4y = 9

2. 4x - 5y = 0

3. 4x + 5y = 40

4. 5x + 4y = 41

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. 2
2. 1
3. 3
4. 0

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

Q. ​​​​​​$\begin{array}{cc}ColumnI& ColumnII\\ \text{a. If}x,y\in R,\text{satisfying the equation}\frac{(x-4{)}^2}{4}+\frac{y^2}{9}=1& \text{p.}-\frac{2}{3}\\ \text{then the difference between the largest and smallest value}& \\ \text{of the expression}\frac{x^2}{4}+\frac{y^2}{9}\text{is}& \\ \text{b. If}PQ\text{is focal chord of ellipse}\frac{x^2}{25}+\frac{y^2}{16}=1\text{which passes}& \text{q.}10\\ \text{through}S\equiv (3,0)\text{and}PS=2,\text{then length of chord}PQ\text{is}& \\ \text{c. If the normal at the point}P\left(\theta \right)\text{to the ellipse}\frac{x^2}{14}+\frac{y^2}{5}=1& \\ \text{intersect it again at the point}Q\left(2\theta \right),\text{then the value of}& \\ \cos \theta \text{is}& \text{r.}\frac{3}{4}\sqrt{7}\\ \text{d. The length of common tangent to}{x}^2+y^2=16\text{and}& \text{s.}8\\ 9x^2+25y^2=225\text{is}& \end{array}$

1. $a-s;b-r;c-q;d-p$
2. $a-p;b-q;c-r;d-s$
3. $a-s;b-q;c-p;d-r$
4. $a-q;b-r;c-s;d-p$

Q. A chord $MP$ parallel to the latus rectum of the ellipse $\frac{x^2}{25}+\frac{y^2}{9}=1$ with centre at $O(0,0)$ intersects the auxiliary circle at $Q$. Then the locus of the point of intersection of normals at $P$ and $Q$ to the respective curve is

1. $x^2+y^2=8$
2. $x^2+y^2=34$
3. $x^2+y^2=64$
4. $x^2+y^2=15$

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