પરવલય y ^ 2 =4 a ( x + a ) નું વિકલ સમીકરણ મેળવો. - Vidyadip

MCQ

પરવલય $y ^{2}=4 a ( x + a )$ નું વિકલ સમીકરણ મેળવો.

A
$y\left(\frac{d y}{d x}\right)^{2}-2 x\left(\frac{d y}{d x}\right)-y=0$
B
$y\left(\frac{d y}{d x}\right)^{2}-2 x\left(\frac{d y}{d x}\right)+y=0$
C
$y\left(\frac{d y}{d x}\right)^{2}+2 x\left(\frac{d y}{d x}\right)-y=0$
D
$y\left(\frac{d y}{d x}\right)+2 x\left(\frac{d y}{d x}\right)-y=0$

✓

Answer

$y ^{2}=4 ax +4 a ^{2}$

differentiate with respect to $x$

$\Rightarrow 2 y \frac{ dy }{ dx }=4 a$

$\Rightarrow a =\left(\frac{ y }{2} \frac{ dy }{ dx }\right)$

so, required differential equation is

$y^{2}=\left(4 \times \frac{y}{2} \frac{d y}{d x}\right) x+4\left(\frac{y}{2} \frac{d y}{d x}\right)^{2}$

$\Rightarrow y ^{2}\left(\frac{ dy }{ dx }\right)^{2}+2 xy \left(\frac{ dy }{ dx }\right)- y ^{2}=0$

$\Rightarrow y \left(\frac{ dy }{ dx }\right)^{2}+2 x \left(\frac{ dy }{ dx }\right)- y =0$

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