Differential Equations
Variable Separable
MJMT_Full_Test_07
Grade 12
Question:
The solution of the differential equation $x\,dy + y\,dx = 0$ passes through the point $(2, 8)$. The latus rectum of the conic represented by the solution curve equals
Step-by-Step Solution
Key Concept: Solve to get $xy = c$; use $(2,8)$ to find $c=16$; identify $xy=16$ as rectangular hyperbola and compute LR.
$x\,dy + y\,dx = d(xy) = 0 \Rightarrow xy = c$. At $(2,8)$: $c=16$, so $xy=16$. This is a rectangular hyperbola with $a^2=16$, LR $= 2a\sqrt{2} = 8\sqrt{2}$.
Correct Answer: 3