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 Question:
We must find the equation of a tangent line. First, we need to locate the specific point where the curve crosses the X-axis. Then, we find the derivative at that point to get the slope, and finally, apply the point-slope form.

Step 2: Detailed Explanation:
1. Find the point of intersection with the X-axis:
A curve crosses the X-axis when $y = 0$.
Substitute $y = 0$ into the curve equation:
$(1 + x^2)(0) = 2 - x$
$0 = 2 - x \implies x = 2$
So, the point of tangency is $(2, 0)$.
2. Find the derivative (slope $m$) using implicit differentiation:
The curve is $(1 + x^2)y = 2 - x$.
Apply the product rule on the left side:
$(2x)y + (1 + x^2)\frac{dy}{dx} = -1$
Substitute the point $(2, 0)$ into this derivative equation:
$(2 \cdot 2)(0) + (1 + 2^2)\frac{dy}{dx} = -1$
$0 + (1 + 4)\frac{dy}{dx} = -1$
$5\frac{dy}{dx} = -1 \implies m = \frac{-1}{5}$
3. Form the equation of the tangent line:
Use the point-slope form: $y - y_1 = m(x - x_1)$
$y - 0 = -\frac{1}{5}(x - 2)$
Multiply both sides by 5:
$5y = -(x - 2)$
$5y = -x + 2$
Rearrange into standard form:
$x + 5y = 2$

Step 3: Final Answer:
The equation of the tangent is $x + 5y = 2$, matching option (a).