Parametric Form of Normal : Ellipse

Q. If the normal at the point $P\left(\theta \right)$ to the ellipse $\frac{x^2}{14}+\frac{y^2}{5}=1$ intersects it again at the point $Q\left(2\theta \right)$, then

1. $\sin \theta =\frac{3}{4}$
2. $\cos \theta =\frac{3}{4}$
3. $\cos \theta =-\frac{2}{3}$
4. $\sin \theta =-\frac{2}{3}$

Q.

The normal to the curve $x=a\left(\cos \left(\theta \right)+\sin \left(\theta \right)\right),y=a\left(\sin \left(\theta \right)-\theta \cos \left(\theta \right)\right)$ at any point $\theta$ is such that

1. It is a constant distance from the origin

2. It passes through $\left(a\frac{\pi }{2},-a\right)$

3. It makes $\frac{\pi }{2}-\theta$ with the $x$-axis

4. It passes through the origin

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. $x+2y=4$

Q. If $CF$ be the perpendicular from the centre $C$ of the ellipse $\frac{x^2}{12}+\frac{y^2}{8}=1,$ on the tangent at any point $P$ and $G$ is the point where the normal at $P$ meets the major axis, then the value of $CF\cdot PG=$

Q. Any ordinate $MP$ of the ellipse $\frac{x^2}{25}+\frac{y^2}{9}=1$ meets the auxiliary circle at $Q$, then locus of the point of intersection of normals at $P$ and $Q$ to the respective curves is

1. $x^2+y^2=8$
2. $x^2+y^2=16$
3. $x^2+y^2=34$
4. $x^2+y^2=64$

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 midpoint of the line segment $PQ$, then the locus of $M$ intersects the latus rectum 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. $(\pm \frac{3\sqrt{5}}{2},\pm \frac{\sqrt{19}}{7})$
4. $(\pm 2\sqrt{3},\pm \frac{1}{7})$

Q. The number of lines that can be drawn through the point $(4,-5)$ at a distance 12 from the point $(-2,3)$ is:

1. 0
2. Infinite
3. 1
4. 2

Q. If normal at a variable point $P$ on the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ of eccentricity $e$ meets the axes of the ellipse at $Q$ and $R,$ then the locus of the midpoint of $QR$ is a conic with an eccentricity $e^{\prime }$ such that

1. $e^{\prime }=e$
2. $e^{\prime }$ is independent of $e$
3. $e^{\prime }=\frac{e}{2}$
4. $e^{\prime }=2e$

Q.

Let $a$ and $b$ be positive real numbers such that $a>1$ and $b<a$. Let $P$ be a point in the first quadrant that lies on the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$. Suppose the tangent to the hyperbola at $P$ passes through the point $\left(1,0\right)$, and suppose the normal to the hyperbola at $P$ cuts off equal intercepts on the coordinate axes. Let $∆$ denote the area of the triangle formed by the tangent at $P$, the normal at $P$ and the x-axis. If $e$ denotes the eccentricity of the hyperbola, then which of the following statements is/are TRUE?

1. $1<e<\sqrt{2}$

2. $\sqrt{2}<e<2$

3. $∆=a^4$

4. $∆=b^4$

Q. If $\beta$ is one of the angles between the normals to the ellipse, $x^2+3y^2=9$ at the points $(3\cos \theta ,\sqrt{3}\sin \theta )$ and $(-3\sin \theta ,\sqrt{3}\cos \theta )$; $\theta \in (0,\frac{\pi }{2});$ then $\frac{2\cot \beta }{\sin 2\theta }$ is equal to

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

Q. The area of the rectangle (in sq. units) formed by the perpendiculars drawn from the centre of the ellipse having major axis and minor axis lengths as $2a$ and $2b$ units respectively to the tangent and normal at a point whose eccentric angle is $\frac{\pi }{4}$ is

1. $\frac{(a^2-b^2)ab}{a^2+b^2}$
2. $\frac{(a^2+b^2)ab}{a^2-b^2}$
3. $\frac{(a^2-b^2)}{ab(a^2+b^2)}$
4. $\frac{(a^2+b^2)}{(a^2-b^2)ab}$

Q. Let $x=4$ be a directrix to an ellipse whose centre is at the origin and its eccentricity is $\frac{1}{2}$. If $P(1,\beta ),\beta >0$ is a point on this ellipse, then the equation of the normal to it at $P$ is

1. $8x-2y=5$
2. $4x-2y=1$
3. $7x-4y=1$
4. $4x-3y=2$

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

1. $\theta ={\cos }^{-1}(-\frac{1}{\sqrt{5}})$
2. $\theta ={\cos }^{-1}\left(\frac{1}{\sqrt{5}}\right)$
3. $t=-\frac{2}{\sqrt{5}}$
4. $t=-\frac{1}{\sqrt{5}}$

Q. $P$ is the point on the ellipse $\frac{x^2}{16}+\frac{y^2}{9}=1$ and $Q$ is the corresponding point on the auxilliary circle of the ellipse. If the line joining centre $C$ to $Q$ meets the normal at $P$ with repsect to the given ellipse at $K,$ then the value of $CK$ is

Q.

The length of the normal to the curve $x=a\left(\theta +\sin \theta \right)$, $y=a\left(1-\cos \theta \right)$ at $\theta =\frac{\mathrm{\pi }}{2}$ is

1. $2a$

2. $\frac{a}{2}$

3. $\frac{a}{\sqrt{2}}$

4. $\sqrt{2}a$

Q. A line $lx+my=n$ is normal to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ under the condition $\frac{n^2}{(a^2-b^2{)}^2}(\frac{a^2}{l^2}+\frac{b^2}{m^2})=k,$ then $7k=$

Q. If the normal at $\theta$ to the ellipse $5x^2+14y^2=70,$ meets the curve again at $2\theta$, then the value of $|3\cos \theta |$ is

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.

Describe parametric equation of a circle

Q. If $Ax+By=5$ is a normal at a point $P$ on the ellipse $\frac{x^2}{9}+\frac{y^2}{4}=1$ whose eccentric angle is $\frac{\pi }{4}$, then the value of $(A+B{)}^2$ is

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. If the latus rectum of the ellipse with axis along 𝑥−axis and centre at origin is 10, distance between foci = length of minor axis, then the equation of the ellipse is $\underset{–}{}$

Q.

The normal of the curve $x=a\left(\cos \theta +\theta \sin \theta \right)$, $y=a\left(\sin \theta -\theta \cos \theta \right)$ at any $\theta$ is such that

1. It makes a constant angle with the $x$-axis

2. It passes through the origin

3. It is at a constant distance from the origin

4. None of the above

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. $\frac{a^2}{l^2}+\frac{b^2}{m^2}=\frac{(a^2+b^2{)}^2}{n^2}$
   
2. $\frac{a^2}{l^2}+\frac{b^2}{m^2}=\frac{(a^2-b^2{)}^2}{n^2}$
   
3. $\frac{a^2}{l^2}+\frac{b^2}{m^2}=\frac{(a^2+b^2{)}^2}{n}$
4. None of these

Q.

If $\sqrt{1-x^6}+\sqrt{1-y^6}=a(x^3-y^3)$ and $\frac{dy}{dx}=f(x,y)\sqrt{\left(\frac{1-y^6}{1-x^6}\right)}$, then

1. $f(x,y)=\frac{y}{x}$

2. $f(x,y)=2\left(\frac{y}{x}\right)$

3. $f(x,y)=\frac{y^2}{x^2}$

4. $f(x,y)=\frac{x^2}{y^2}$

Q. If a tangent on ellipse at $A(1,1)$ intersects its directrix at $B(7,–7)$ and $S$ be the corresponding focus of ellipse and $C(\alpha ,\beta )$ is the circumcentre of $△SAB,$ then

1. $\alpha +\beta =1$
2. $\alpha -\beta =7$
3. $S{C}^2=20.5$
4. $S{C}^2=25$

Q. $\cos \left({\sin }^{-1}\left(\tan \left({\cos }^{-1}\left(\sin \left({\tan }^{-1}\frac{4}{3}\right)\right)\right)\right)\right)$ is equal to

1. $\frac{3}{5}$
2. $\frac{\sqrt{7}}{4}$
3. $\frac{3}{4}$
4. $\frac{4}{5}$

Q. Circles $C_1$ and $C_2$ , of radii $r$ and $R$ respectively, touch each other as shown in the figure. The line $A$, which is parallel to the line joining the centres of $C_1$ and $C_2$ , is tangent to $C_1$ at $P$ and intersects $C_2$ at $A,B$. If $R^2=2r^2$, then $\angle AOB$ equals

1. $22{\frac{1}{2}}^{\circ }$
2. ${45}^{\circ }$
3. ${60}^{\circ }$
4. $67{\frac{1}{2}}^{\circ }$

Q. An ellipse whose axis is along $x$ axis and centre at origin. If the distance between focii is equal to length of minor axis, then the eccentricity of the ellipse is

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