The equations of the latusrectum of the ellipse $9 x^2+4 y^2-18 x-8 y-23=0$ are

The equations of the latusrectum of the ellipse $9 x^2+4 y^2-18 x-8 y-23=0$ are
  1. $x=-1 \pm \sqrt{5}$
  2. $y=1 \pm \sqrt{5}$
  3. $x=1 \pm \frac{2 \sqrt{5}}{3}$
  4. $y=2 \pm \sqrt{5}$

Solution

Equation of given ellipse $ \begin{aligned} & 9 x^2+4 y^2-18 x-8 y-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 \\ & \end{aligned} $ Equation of latus rectum is $ \begin{array}{lrl} \Rightarrow & y-1 & = \pm 3 \frac{\sqrt{5}}{3} \\ \Rightarrow & y & =1 \pm \sqrt{5} \end{array} $

Asked in: AP EAMCET 2018 (22 Apr Shift 2)

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