MCQ
If $u = {\tan ^{ - 1}}(x + y),$ then $x{{\partial u} \over {\partial x}} + y{{\partial u} \over {\partial y}} = $
- A$\sin 2u$
- ✓${1 \over 2}\sin 2u$
- C$2\tan u$
- D${\sec ^2}u$
$\therefore $ $\tan u$ is homogeneous in $x,\,y$ of order $ 1$.
$\therefore $ $x\frac{\partial }{{\partial x}}(\tan u) + y\frac{\partial }{{\partial y}}(\tan u) = \tan u$
$\therefore $ $x{\sec ^2}u\frac{{\partial u}}{{\partial x}} + y{\sec ^2}u\frac{{\partial u}}{{\partial y}} = \tan u$
$\therefore $ $x\frac{{\partial u}}{{\partial x}} + y\frac{{\partial u}}{{\partial y}} = \tan u.{\cos ^2}u = \sin u\cos u$ = $\frac{1}{2}\sin 2u$.
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
$(A)$ The triangle $PFQ$ is a right-angled triangle
$(B)$ The triangle $QPQ ^{\prime}$ is a right-angled triangle
$(C)$ The distance between $P$ and $F$ is $5 \sqrt{2}$
$(D)$ $F$ lies on the line joining $Q$ and $Q ^{\prime}$