Question

In a rectangular hyperbola, C is the centre of hyperbola. Normal at any point P meet the axes in G and g, then

• A. PG = Pg
• B. PG = PC
• C. Pg = PC
• D. Pg = 2PC

Answer 1

Answer: (A), (B), (C)

The correct options are
A PG = Pg
B PG = PC
C Pg = PC

Let the rectangular hyperbola be $x^2-y^2=a^2$

Let point $P(a\sec \varphi ,a\tan \varphi )$

Equation of normal at P; $x\sin \varphi +y=2a\tan \varphi$

Intersection with $y=0$ gives $x=2a\sec \varphi$

Intersection with $x=0$ gives $y=2a\tan \varphi$



$CG=2a\sec \varphi$

$Cg=2a\tan \varphi$

$P{G}^2=(a\sec \varphi -2a\sec \varphi {)}^2+(a\tan \varphi -0{)}^2$

$=a^2({\sec }^2\varphi +{\tan }^2\varphi )$

$P{g}^2=(a\sec \varphi -0{)}^2+(a\tan \varphi -2a\tan \varphi {)}^2$

$=a^2({\sec }^2\varphi +{\tan }^2\varphi )$

$P{C}^2=(a\sec \varphi -0{)}^2+(a\tan \varphi -0{)}^2$

$=a^2({\sec }^2\varphi +{\tan }^2\varphi )$

$\Rightarrow PG=Pg=PC$