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

The Correct Option is A

To find the value of \( A \) where the curve \( y(1 + x^2) = 2 - x \) crosses the x-axis, we set \( y = 0 \). This gives:

\[ 0 \cdot (1 + x^2) = 2 - x \]

Thus, \( 2 - x = 0 \) implies \( x = 2 \).

At this point, substitute \( x = 2 \) into the original equation:

\[ y(1 + 2^2) = 2 - 2 \]

\[ y \cdot 5 = 0 \]

\[ y = 0 \]

So, the point where the curve crosses the x-axis is \( (2, 0) \).

Next, we differentiate the given equation implicitly with respect to \( x \):

\[ \frac{d}{dx}[y(1 + x^2)] = \frac{d}{dx}(2 - x) \]

Using the product rule, we have:

\[ (1 + x^2)\frac{dy}{dx} + y \cdot \frac{d}{dx}(1 + x^2) = -1 \]

\[ (1 + x^2)\frac{dy}{dx} + 2xy = -1 \]

Substituting \( (x, y) = (2, 0) \) into the differentiated equation:

\[ (1 + 2^2)\frac{dy}{dx} + 2 \cdot 2 \cdot 0 = -1 \]

\[ 5\frac{dy}{dx} = -1 \]

Thus,

\[ \frac{dy}{dx} = -\frac{1}{5} \]

Since \(\frac{dy}{dx} = \frac{1}{A}\), we equate:

\[-\frac{1}{5} = \frac{1}{A} \]

Solve for \( A \):

\[ A = -5 \]

The value of \( A \) is -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:
$-5$