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The vertices of the ellipse \mathrm{9 x^2+4 y^2-18 x-27=0} are \mathrm{(\alpha, \pm \beta)} then value of \mathrm{|\alpha|+|\beta|} is _________.

Option: 1

4


Option: 2

5


Option: 3

6


Option: 4

7


Answers (1)

best_answer

The equation of the ellipse can be written as

\begin{aligned} & 9\left(x^2-2 x+1\right)+4 y^2=36 \\ & \Rightarrow 9(x-1)^2+4(y-0)^2=36 \\ & \Rightarrow \frac{(x-1)^2}{2^2}+\frac{(y-0)^2}{3^2}=1 \end{aligned}
Shifting the origin at (1,0), keeping the axes parallel to the coordinate axes, we have

x=X+1, y=Y+0                                    ...(i)
The equation of the ellipse with reference to new origin is

\frac{X^2}{2^2}+\frac{Y^2}{3^2}=1
The coordinates of vertices with reference to new origin are

 (X=0, Y= \pm 3)
Substituting these in (i), we obtain (1, \pm 3) as the coordinates of the vertices of the given conic.
Compare with (\alpha, \pm \beta)

|\alpha|+|\beta|=1+3=4

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