Question

Which among the following point lie inside the hyperbola $\frac{x^2}{3}-\frac{y^2}{5}=1$

• A. $(3,1)$
• B. $(5,3)$
• C. $(4,5)$
• D. $(1,0)$

Answer 1

Answer: (A), (B)

The correct options are
A $(3,1)$
B $(5,3)$

Condition for a point to lie inside hyperbola is $S_1>0$

$\left(a\right)\frac{9}{3}-\frac{1}{5}-1>0\Rightarrow 3-\frac{6}{5}>0\Rightarrow \frac{15-6}{5}>0$
lies inside hyperbola

$\left(b\right)\frac{25}{3}-\frac{9}{5}-1>0\Rightarrow \frac{25\times 5-27-15}{15}>0\Rightarrow \frac{125-42}{15}>0$
lies inside hyperbola

$\left(c\right)\frac{16}{3}-\frac{25}{5}-1>0\Rightarrow \frac{16}{3}-6>0\Rightarrow \frac{16-18}{3}<0$
lies outside hyperbola

$\left(d\right)\frac{1}{3}-1<0$
lies outside hyperbola