Question

\[\frac{x^{2}}{r^{2}-r-6}+\frac{y^{2}}{r^{2}-6r+5}=1\] will represent the ellipse, if r lies in the interval______.

Options

• \[(-\infty,2)\]

• \[(3,\infty)\]

• \[(5,\infty)\]

• \[(1,\infty)\]

Answer 1

\[\frac{x^{2}}{r^{2}-r-6}+\frac{y^{2}}{r^{2}-6r+5}=1\] will represent the ellipse, if r lies in the interval \[(5,\infty)\].

Explanation:

\[r^{2}-r-6>0\] and \[r^{2}-6r+5>0\]

\[\Rightarrow\quad(r-3)(r+2)>0\]

and \[\begin{pmatrix} r-1 \end{pmatrix} \begin{pmatrix} r-5 \end{pmatrix}>0\]

\[\Rightarrow\quad\{r<-2\mathrm{or}r>3\}\cap\{r<1\mathrm{or}r>5\}\]

\[\Rightarrow\quad r<-2\quad\mathrm{or}\quad r>5\]