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The new equation of the curve $4 (x - 2 y + 1)^{2} + 9 (2 x + y + 2)^{2} = 25,$ if the lines $2 x + y + 2 = 0$ and $x - 2 y + 1 = 0$ are taken as the new x and y axes respectively is

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$4 x^{2} + 9 y^{2} = 5$

$4 x^{2} + 9 y^{2} = 7$

$4 x^{2} + 9 y^{2} = 25$

$4 x^{2} + 9 y^{2} = 9$

answer is A.

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Detailed Solution

$a x + b y + c = 0, b x - a y + k = 0$ takes new x,y axis then $x = \frac{b x - a y + k}{\sqrt{a^{2} + b^{2}}}$

$y = \frac{a x + b y + c}{\sqrt{a^{2} + b^{2}}}$

$T . E is 4 {(\frac{x - 2 y + 1}{\sqrt{5}})}^{2} + 9 {(\frac{2 x + y + 2}{\sqrt{5}})}^{2} = 5$

$4 x^{2} + 9 y^{2} = 5$

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