Question

The equation of the curve passing through $(2,\frac{7}{2})$ and having gradient $1-\frac{1}{x^2}$ at a point $(x,y)$ is:

• A. $y=x^2+x+1$
• B. $xy=x^2+x+1$
• C. $xy=x+5$
• D. $xy=x^2+3$

Answer 1

The correct option is B $xy=x^2+x+1$

We have,
$\frac{dy}{dx}=1-\frac{1}{x^2}\Rightarrow dy=(1-\frac{1}{x^2})dx$
On integrating both sides we have:
$\Rightarrow y=x+\frac{1}{x}+C$
This passes through $(2,\frac{7}{2})$,
$\Rightarrow \frac{7}{2}=2+\frac{1}{2}+C\Rightarrow C=1$
Thus the equation of the curve is
$y=x+\frac{1}{x}+1\Rightarrow xy=x^2+x+1$