Question

The equation of the latus rectum of the ellipse $9x^2+4y^2-18x-8y-23=0$ are

• A. $y=\pm \sqrt{5}$
• B. $y=-\sqrt{5}$
• C. $y=1\pm \sqrt{5}$
• D. $y=-1\pm \sqrt{5}$

Answer 1

The correct option is C $y=1\pm \sqrt{5}$

$9x^2+4y^2-18x-8y-23=0$

$=(3x-3{)}^2+(2y-2{)}^2-13-23=0$

$\Rightarrow 9(x-1{)}^2+4(y-1{)}^2=36$

$\Rightarrow \frac{(x-1{)}^2}{4}+\frac{(y-1{)}^2}{9}=1$

Shifting origin to $(1,1)\Rightarrow x-1=x,y-1=y$

$\Rightarrow \frac{x^2}{4}+\frac{y^2}{9}=1$

$a=2,=b=3,e^2=1+\frac{a^2}{b^2}=1+\frac{4}{9}=\frac{5}{9}$

$\Rightarrow e=\pm \sqrt{\frac{5}{9}}+\frac{\sqrt{5}}{3}$

$\Rightarrow$ Focus $=(0,\pm be)=(0,\pm \sqrt{5})$

$\Rightarrow$ latus ractum $\Rightarrow y=\pm \sqrt{5}$

Shifting back, $y=y-1$

$\Rightarrow y-1=\pm \sqrt{5}$

$\Rightarrow y=1\pm \sqrt{5}\Rightarrow \left(C\right)$