Parametric Form of Normal : Hyperbola

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

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