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  4. Latus rectum of the conic satisfying the differential equation, x d y+y d x=0 and passing through the point (2,8) is :

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Q. Latus rectum of the conic satisfying the differential equation, $x d y+y d x=0$ and passing through the point $(2,8)$ is :

Conic Sections

Solution:

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$\frac{ dy }{ y }+\frac{ dx }{ x }=0 \Rightarrow \ln xy = c \Rightarrow xy = c $
$\text { passes through }(2,8) \Rightarrow c =16$
$xy =16 LR =2 a \left( e ^2-1\right)=2 a ( e =\sqrt{2})$
$\text { solving with } y = x$
$\text { vertex is }(4,4) $
$\text { distance from centre to vertex }=4 \sqrt{2}$
L.R. $=$ length of $TA =8 \sqrt{2}$