Question

If the tangent to the curve $xy+ax+by=0$ at (1, 1) makes an angle ${\tan }^{-1}\left(2\right)$ with x-axis, then $a+2b$ is equal to

• A. $\frac{1}{2}$
• B. $-\frac{1}{2}$
• C. $3$
• D. $-3$

Answer 1

The correct option is D $-3$

$x{y}^{\prime }+y+a+b{y}^{\prime }=0$
$(x+b)y^{\prime }+a+y=0$
$y^{\prime }=\frac{-(a+y)}{x+b}$
Now
$y_{1,1}^{\prime }=\frac{-(a+1)}{b+1}$
$=2$
Or
$2b+2=-a-1$
$2b+a=-3$