Question

For the curve $y(1 + x^2) = 2 - x$, if $\frac{dy}{dx} = \frac{1}{A}$ at the point where the curve crosses the x-axis, then the value of $A$ is:

• A. -5
• B. 5
• C. -1
• D. 0

Answer 1

Answer: (A)

-5

Answer 2

The equation of the curve is:
$y(1 + x^2) = 2 - x$.
Step 1: Find the point where the curve crosses the $x$-axis.
At the $x$-axis, $y = 0$. Substitute $y = 0$ into the equation:
$0(1 + x^2) = 2 - x \implies x = 2$.
Thus, the curve crosses the $x$-axis at $(2, 0)$.
Step 2: Differentiate the equation.
Differentiate $y(1 + x^2) = 2 - x$ using the product rule:
$\frac{d}{dx} y(1 + x^2) = \frac{d}{dx} (2 - x)$
$y \cdot \frac{d}{dx}(1 + x^2) + (1 + x^2) \cdot \frac{dy}{dx} = -1$.
$y(2x) + (1 + x^2) \frac{dy}{dx} = -1$.
Step 3: Evaluate at $(x, y) = (2, 0)$.
Substitute $x = 2$ and $y = 0$ into the differentiated equation:
$0(2 \cdot 2) + (1 + 2^2) \frac{dy}{dx} = -1$.
$(1 + 4) \frac{dy}{dx} = -1 \implies \frac{dy}{dx} = -1/5$.
Step 4: Relate $\frac{dy}{dx}$ to $A$.
It is given that:
$\frac{dy}{dx} = \frac{1}{A}$
Equating:
$\frac{1}{5} = \frac{1}{A}$
Solve for $A$:
$A = -5$.
Final Answer:
$-5$