Parametric Form of Normal : Hyperbola

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