Question

If the straight line $y-2x+1=0$ is the tangent to the curve $xy+ax+by=0$ at $x=1,$ then the values of $a$ and $b$ are respectively :

• A. 1 and 2
• B. 1 and -1
• C. -1 and 2
• D. -1 and -2
• E. 1 and -2

Answer 1

The correct option is E 1 and -2

Equation of tangent is $y-2x+1=0$

It is tangent at $x=1$, so for $x=1$

$y-2\left(1\right)+1=0\Rightarrow y=1$

So, its is tangent to the curve at $(1,1)$

Slope of tangent $=-\left(\frac{-2}{1}\right)=2$

$xy+ax+by=0y+x\frac{dy}{dx}+a+b\frac{dy}{dx}=0$

Now $\frac{dy}{dx}=2$ at $(1,1)$

$1+1\left(2\right)+a+b\left(2\right)=0\Rightarrow a+2b=-3$ .....(i)

$(1,1)$ also lies on the curve $xy+ax+by=0$

$\Rightarrow 1\left(1\right)+a\left(1\right)+b\left(1\right)=0\Rightarrow a+b=-1$ .......(ii)

Solving (i) and (ii), we get

$\Rightarrow a=1,b=-2$

So, option E is correct.