Parametric Form of Normal : Hyperbola

Q.

The equation of normal to a hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ at a point with parameter $\theta$ is

$axcos\theta -bycot\theta =a^2+b^2$

Q.

Differential coefficient of ${\sec }^{-1}\left(\frac{1}{2x^2-1}\right)$ with respect to $\sqrt{\left(1-x^2\right)}$ at $x=\frac{1}{2}$ is

1. $2$

2. $4$

3. $6$

4. $1$

Q.

Difference of slopes of the lines represented by equation $x^2\left({\sec }^2\left(\theta \right)-{\sin }^2\left(\theta \right)\right)-2xy\tan \left(\theta \right)+y^2{\sin }^2\left(\theta \right)=0$ is

1. $4$

2. $3$

3. $2$

4. none of these

Q. If the normal at any point P on the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ meets the co-ordinate axes in G and g respectively, then PG : Pg =

1. $a:b$
2. $a^2:b^2$
3. $b^2:a^2$
4. $b:a$

Q. A normal to the hyperbola $x^2-4y^2=4$ meets the $x$ and $y$ axes at $A$ and $B$ respectively. The locus of point of intersection of the straight lines drawn through $A$ and $B,$ perpendicular to the $x$ and $y$ axes respectively is a hyperbola with

1. eccentricity equal to $\sqrt{3}$
2. length of latus rectum equal to $20$
3. equation of auxiliary circle equal to $x^2+y^2=25$
4. distance between foci equal to $5\sqrt{5}$

Q. The condition that the straight line $lx+my=n$ may be a normal to the hyperbola $b^2{x}^2-a^2{y}^2=a^2{b}^2$ is given by

1. $\frac{a^2}{l^2}-\frac{b^2}{m^2}=\frac{(a^2+b^2{)}^2}{n^2}$
2. $\frac{l^2}{a^2}-\frac{m^2}{b^2}=\frac{(a^2+b^2{)}^2}{n^2}$
3. $\frac{a^2}{l^2}+\frac{b^2}{m^2}=\frac{(a^2-b^2{)}^2}{n^2}$
4. $\frac{l^2}{b^2}+\frac{m^2}{b^2}=\frac{(a^2-b^2{)}^2}{n^2}$

Q. The length of the chord of the parabola $y^2=4ax(a>0)$ which passes through the vertex and makes an acute angle $\alpha$ with the axis of the parabola is

1. $\pm 4a\cot \alpha \text{cosec}\alpha$
2. $4a\cot \alpha \text{cosec}\alpha$
3. $-4a\cot \alpha \text{cosec}\alpha$
4. $4a{\text{cosec}}^2\alpha$

Q.

If $p$ and $p$ be the distances of origin from the lines$xsec\alpha +y\cos ec\alpha =k$ and$x\cos \alpha -y\sin \alpha =k\cos 2\alpha$, then $4p^2+p{}^2$

1. $k$

2. $2k$

3. $k^2$

4. $2k^2$

Q. If the normal at angle $\varphi$ on the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ meets the transverse axis at $G$ such that $AG\cdot A^{\prime }G=a^m(e^n{\sec }^p\varphi -1)$, where $A,A^{\prime }$ are the vertices of the hyperbola, $e$ is the eccentricity and $m,n$ and $p$ are positive integers, then value of $(m+n+p)$ is

Q. Let $P(a\sec \theta ,b\tan \theta )$ and $Q(a\sec \varphi ,b\tan \varphi ),$ where $\theta +\varphi =\frac{\pi }{2}$, be two points on the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$. If $(h,k)$ is the point of intersection of normals at $P$ and $Q$, then $k$ is equal to

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

Q.

Compute the area of the curvilinear triangle bounded by the $y$-axis and the curves, $y=\tan x$ and $y=\frac{2}{3}\cos x$.

Q. If the distance between two parallel tangents drawn to the hyperbola $\frac{x^2}{9}-\frac{y^2}{49}=1$ is $2$, then their slope is equal to

1. $\pm \frac{3}{2}$
2. $\pm \frac{5}{2}$
3. None of these
4. $\pm \frac{4}{5}$

Q.

If $ST$ and $SN$ are the lengths of the subtangent and the subnormal at the point $\theta =\frac{\pi }{2}$ on the curve$x=a(\theta +\sin \theta ),y=a(1-\cos \theta ),a\ne 1,$ then

1. $ST=SN$

2. $ST=2SN$

3. $ST2=aS{N}^3$

4. $S{T}^3=aSN$

Q. The normal at $P$ to the hyperbola $\frac{x^2}{9}-\frac{y^2}{1}=1$ meets the transverse axis $A{A}^{\prime }$ at $G$ and the conjugate axis $B{B}^{\prime }$ at $H.$ If $CF$ is the perpendicular drawn from centre $C$ of the hyperbola to the normal, then

1. $PF\cdot PG=C{B}^2$
2. $PF\cdot PG=2C{B}^2$
3. Locus of mid-point of $GH$ is another hyperbola with eccentricity $=\sqrt{10}$
4. $PF\cdot PH=C{A}^2$

Q. Equation of normal to hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ at (a sec $\theta$, b tan $\theta$) is ax cos $\theta$ - by cot $\theta$ = $a^2+b^2.$

1. True
2. False

Q.

A normal is drawn on the hyperbola $\frac{x^2}{16}-\frac{y^2}{9}=1$ at a point given by parameter $\frac{\pi }{3}$.

What is the x-intercept of the normal.

1. 

2. 

3. 

4. 

Q.

What is the eccentricity of a hyperbola if a chord joining 2 points given by parameters $\alpha$ and $\beta$ also passes through the focus. [Given :$\alpha +\beta =p,\alpha -\beta =q]$

1. cos

2. 

3. 

4. cos

Q. The condition that the straight line $lx+my=n$ may be a normal to the hyperbola $b^2{x}^2-a^2{y}^2=a^2{b}^2$ is given by

1. $\frac{a^2}{l^2}-\frac{b^2}{m^2}=\frac{(a^2+b^2{)}^2}{n^2}$
2. $\frac{l^2}{a^2}-\frac{m^2}{b^2}=\frac{(a^2+b^2{)}^2}{n^2}$
3. $\frac{a^2}{l^2}+\frac{b^2}{m^2}=\frac{(a^2-b^2{)}^2}{n^2}$
4. $\frac{l^2}{b^2}+\frac{m^2}{b^2}=\frac{(a^2-b^2{)}^2}{n^2}$

Q. Let $P(a\sec \theta ,b\tan \theta )$ and $Q(a\sec \varphi ,b\tan \varphi ),$ where $\theta +\varphi =\frac{\pi }{2}$, be two points on the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$. If $(h,k)$ is the point of intersection of normals at $P$ and $Q$, then $k$ is equal to

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

Q. Equation of normal to hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ at (a sec $\theta$, b tan $\theta$) is ax cos $\theta$ - by cot $\theta$ = $a^2+b^2.$

1. True
2. False

Q. Find the maximum area of an isosceles triangle inscribed in the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ with is vertex as at on end of the major axis.

Q. In the given figure vertices of $△ABC$ lie on y = $f\left(x\right)=a{x}^2+bx+c$. (vertex lies on y axis). The $△ABC$ is right angled isosceles triangle whose hypotenuse $AC=8units$, then

1. $y=f\left(x\right)=\frac{x^2}{4}-4$
2. minimum value of f(x) is $-4$
3. Number of integral values of x lying between the roots of f(x) = 0 are $7$
4. Area of the $△ABC$ is $16$ sq.units

Q. Tangent to parabola $y^2=4x+5$ which is parallel to $y=2x+7$

1. $y-2x-3=0$
2. $y-2x+1=0$
3. $y=x+1$
4. $y=x+3$

Q. A normal to the hyperbola $\frac{x^2}{4}-\frac{y^2}{1}=1$ has equal intercepts on positive $x$ and $y$ axes. If this normal touches the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, then $a^2+b^2$ is equal to

1. $25$
2. $5$
3. $16$
4. None of these

Q. Let $\stackrel{\_}{v}$ be a unit vector which follows the equation, $\stackrel{\_}{v}\times \stackrel{\_}{b}=\stackrel{\_}{c}$. also $\left|\stackrel{\_}{b}\right|=2$ and $\left|\stackrel{\_}{c}\right|=\sqrt{3}$ then

1. $\stackrel{\_}{v=-}\stackrel{\_}{b}+\stackrel{\_}{b}\times \stackrel{\_}{c}$
2. $\stackrel{\_}{v}=\frac{3}{4}(\stackrel{\_}{b}+2\stackrel{\_}{b}\times \stackrel{\_}{c})$
3. none of these
4. $\stackrel{\_}{v}=\frac{1}{4}(\stackrel{\_}{b}+\stackrel{\_}{b}\times \stackrel{\_}{c})$

Q.

Find the equation and the length of the common tangents to hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ and $\frac{y^2}{a^2}-\frac{x^2}{b^2}=1$

Q. The normal at $P$ to the hyperbola $\frac{x^2}{9}-\frac{y^2}{1}=1$ meets the transverse axis $A{A}^{\prime }$ at $G$ and the conjugate axis $B{B}^{\prime }$ at $H.$ If $CF$ is the perpendicular drawn from centre $C$ of the hyperbola to the normal, then

1. $PF\cdot PG=C{B}^2$
2. $PF\cdot PG=2C{B}^2$
3. Locus of mid-point of $GH$ is another hyperbola with eccentricity $=\sqrt{10}$
4. $PF\cdot PH=C{A}^2$

Q. If $\tan \theta =\sqrt{3}$ then the number of solutions for $\theta$ if $0\le \theta \le {360}^0$ is:

Q. If the normal at $P$ to the rectangular hyperbola $x^2-y^2=4$ meeets the axes in $G$ and $g$ and $C$ is the centre of the hyperbola, then

1. $PG=PC$
2. $Pg=PC$
3. $PG=Pg$
4. $Gg=2PC$

Q. If $2sec\theta =4$, then the value of $\theta$ is

1. ${30}^{\circ }$
2. ${45}^{\circ }$
3. ${60}^{\circ }$
4. ${90}^{\circ }$