Question

Equation of the curve passing through the point $(4,3)$ and having slope $=y/2$ at a point $(x,y)$ on it is

• A. $\log \left(\frac{y}{3}\right)=\frac{x}{2}-2$
• B. $\log \left(\frac{y}{3}\right)=\frac{x}{2}+2$
• C. $y=3e^{x-2}$
• D. $y^2=x+5$

Answer 1

The correct option is A $\log \left(\frac{y}{3}\right)=\frac{x}{2}-2$

$\frac{dy}{dx}=\frac{y}{2}$ (given)

$\Rightarrow \int \frac{dy}{dx}=\int \frac{dx}{2}$

$\Rightarrow \log y=\frac{1}{2}x+c$

since this curve passes through $\left(4,3\right)$

$\log 3=\frac{1}{2}\times 4+c$

$\Rightarrow \log 3=2+c$

$\Rightarrow c=\log 3-2$

Hence Equation of the curve is

$\log y=\frac{1}{2}x+\log 3-2$

$\Rightarrow \log y-\log 3=\frac{x}{2}-2$

$\Rightarrow \log \left(\frac{y}{3}\right)=\frac{x}{2}-2$