Parametric Form of Normal : Ellipse

Q. The line lx + my + n = 0 will be a normal to the hyperbola $b^2{x}^2-a^2{y}^2=a^2{b}^2$, if

1. 
   
2. 
   
3. 
4. None of these

Q. The normal at $P\left(\theta \right)$ and D$(\theta +\frac{\pi }{2})$ meet the major axis of $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ at Q and R. Then $P{Q}^2+D{R}^2=$

1. 
2. 
3. 
4. 

Q. If the normal at $\theta$ on the ellipse $5x^2+14y^2=70$ cuts the curve again at a point $2\theta$, then $\cos \theta$ =

1. 
2. 
3. 
4. 

Q.

The normal at a point P on the ellipse $x^2+4y^2=16$ meets the X - axis at Q. If M is the mid-point of the line segment PQ, then the locus of M intersects the latusrectum of the given ellipse at the points

1. $(\pm \frac{3\sqrt{5}}{2},\pm \frac{2}{7})$

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

3. $(\pm \frac{3\sqrt{5}}{2},\pm \frac{\sqrt{19}}{4})$

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

Q. If the normal to the ellipse $3x^2+4y^2=12$ at a point $P$ on it is parallel to the line, $2x+y=4$ and the tangent to the ellipse at $P$ passes through $Q(4,4)$ then $PQ$ is equal to

1. $\frac{5\sqrt{5}}{2}$ unit
2. $\frac{\sqrt{157}}{2}$ unit
3. $\frac{\sqrt{221}}{2}$ unit
4. $\frac{\sqrt{61}}{2}$ unit

Q. If the normal at any point $P$ on the ellipse $\frac{x^2}{9}+\frac{y^2}{16}=1$ meets the axes at $G$ and $g,$ respectively, then the ratio of $PG:Pg$ is

1. $3:4$
2. $4:3$
3. $9:16$
4. $16:9$

Q. If the tangent drawn at point $P(t^2,2t)$ on the parabola $y^2=4x$ is same as the normal drawn at point $Q(\sqrt{5}\cos \theta ,2\sin \theta )$ on the ellipse $4x^2+5y^2=20,$ then

1. $Q\equiv (-1,\frac{4}{\sqrt{5}})$
2. $Q\equiv (-1,-\frac{4}{\sqrt{5}})$
3. $P\equiv (\frac{1}{5},-\frac{2}{\sqrt{5}})$
4. $P\equiv (\frac{1}{5},\frac{2}{\sqrt{5}})$

Q. Which of the following is/are true ?

1. There are infinite positive integral values of $a$ for which $(13x-1{)}^2+(13y-2{)}^2={\left(\frac{5x+12y-1}{a}\right)}^2$ represents an ellipse
2. The minimum distance of a point $(1,2)$ from the ellipse $4x^2+9y^2+8x-36y+4=0$ is $1$ unit
3. If from a point $P(0,\alpha )$ two normals other than axes are drawn to the ellipse $\frac{x^2}{25}+\frac{y^2}{16}=1$, then $|\alpha |<\frac{9}{4}$
4. If the length of latus rectum of an ellipse is one-third of its major axis, then its eccentricity is equal to $\frac{1}{\sqrt{3}}$

Q. Sum of distance's from the $x-$axis to the point(s) on the ellipse $\frac{x^2}{9}+\frac{y^2}{4}=1,$ where the normal is parallel to the line $2x+y=1,$ is $\frac{k}{5}$ unit, then $k=$

Q. The eccentricity of an ellipse whose centre is 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. $x+2y=4$