Question

The equation $\frac{x^2}{8-t}+\frac{y^2}{t-4}=1$ will represent an ellipse if

• A. $t\in (1,5)$
• B. $t\in (2,8)$
• C. $t\in (4,8)-\left\{6\right\}$
• D. $t\in (4,10)-\left\{6\right\}$

Answer 1

The correct option is C $t\in (4,8)-\left\{6\right\}$

Consider Equation, $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ to represent an ellipse equation.
$a>0,b>0,a\ne b$
Given,equation $\frac{x^2}{(8-t)}+\frac{y^2}{(t-4)}=1$
$\Rightarrow (8-t)>0$ and $(t-4)>0,(8-t)\ne (t-4)$
$\Rightarrow t\in (-\infty ,8)\cap (4,\infty )\cap$ {$t\ne 6$}
$\Rightarrow t\in (4,8)-${$6$}