Q96 / 160 · AP EAMCET 2006 · Mathematics

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

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

Solution

Equation of ellipse is $9 x^2+4 y^2-18 x-8 y-23=0$ $\Rightarrow \quad 9\left(x^2-2 x+1\right)-9+4\left(y^2-2 y+1\right)$ $-4-23=0$ $\Rightarrow \quad 9(x-1)^2+4(y-1)^2=36$ $\Rightarrow \quad \frac{(x-1)^2}{4}+\frac{(y-1)^2}{9}=1$ Here $\quad a < b$ Also$e=\sqrt{1-\frac{4}{9}}=\sqrt{\frac{5}{9}}=\frac{\sqrt{5}}{3}$ Equation of latus rectum are $y-1= \pm 3 \cdot \frac{\sqrt{5}}{3}$ $y=1 \pm \sqrt{5}$

Hence, the correct answer is option: 'C'.

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