The following equation represents an ellipse, $25 (x^{2} - 6 x + 9) + 16 y^{2} = 400$ . How should the axes be transformed so that the ellipse is represented by the equation $\frac{x^{2}}{25} + \frac{y^{2}}{16} = 1.$

the origin should be shifted to the point

(3, 0)

and then the axes be turned through a right angle.

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the origin should be shifted to the point

(- 3, 0)

and then the axes be turned through a right angle.

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the origin should be shifted to the point

(0, 3)

and then the axes be turned through a right angle.

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the origin should be shifted to the point

(0, - 3)

and then the axes be turned through a right angle.

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Solution

The correct option is A the origin should be shifted to the point $(3, 0)$ and then the axes be turned through a right angle.
$25 (x^{2} - 6 x + 9) + 16 y^{2} = 400. O r, 25 {(x - 3)}^{2} + 16 y^{2} = 400 O r, \frac{{(x - 3)}^{2}}{16} + \frac{y^{2}}{25} = 1 O r, \frac{X^{2}}{16} + \frac{Y^{2}}{25} = 1 X = x - 3 Y = 0 N o w i f w e t u r n t h e a x i s t h r o u g h a r i g h t a n g l e, \frac{X^{2}}{25} + \frac{Y^{2}}{16} = 1 ∴ T h e o r i g i n s h o u l d b e s h i f t e d t o (3, 0) a n d t h e n t h e a x i s b e t u r n e d t h r o u g h r i g h t a n g l e .$

Option [A]

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