Question

The equation of the tangent to the curve $(1 + x^2)y = 2 - x$, where it crosses the X-axis, is ______.

• A. $x + 5y = 2$
• B. $x - 5y = 2$
• C. $5x - y = 10$
• D. $5x + y - 10 = 0$

Hint:

Using implicit differentiation is often much faster than isolating $y$! By leaving the function as $(1+x^2)y = 2-x$, plugging in $y=0$ instantly annihilates entire massive terms during the derivative calculation.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
A curve crosses the X-axis when $y = 0$. We need to find this point and the slope ($dy/dx$) at that point.

Step 2: Formula Application:
$(1 + x^2)y = 2 - x$. Setting $y = 0$: $0 = 2 - x \implies x = 2$. Point is $(2, 0)$.

Step 3: Explanation:
Differentiating: $(1+x^2)y' + 2xy = -1$. At $(2, 0)$: $(1 + 2^2)y' + 2(2)(0) = -1 \implies 5y' = -1 \implies y' = -1/5$. Equation: $y - 0 = -\frac{1}{5}(x - 2) \implies 5y = -x + 2 \implies x + 5y = 2$.

Step 4: Final Answer:
The equation is $x + 5y = 2$.