Verified Answer :
- Verify that the point \((3, -1)\) lies on the curve \(x^2 - 2xy = 15\).
- Apply implicit differentiation to find the derivative \(\frac{dy}{dx} = \frac{x - y}{x}\).
- Calculate the slope of the tangent at \((3, -1)\) by substituting the coordinates into the derivative, obtaining \(m = \frac{4}{3}\).
- Use the point-slope form to determine the tangent line equation, resulting in \(\boxed{y = \frac{4}{3}x - 5}\).
Explanation
- Problem Analysis and Verification
To find the equation of the tangent line to the curve defined by \(x^2 - 2xy = 15\) at the point \((3, -1)\), we first verify that the point lies on the curve. Substituting \(x = 3\) and \(y = -1\) into the equation, we get \(3^2 - 2(3)(-1) = 9 + 6 = 15\). This confirms the point \((3, -1)\) is indeed on the curve. We will now find the derivative \(\frac{dy}{dx}\) using implicit differentiation.
- Implicit Differentiation
We differentiate both sides of \(x^2 - 2xy = 15\) with respect to \(x\).
Applying the power rule to \(x^2\) gives \(2x\).
For the term \(-2xy\), we use the product rule: \(\frac{d}{dx}(-2xy) = -2(y + x\frac{dy}{dx})\).
Setting the derivative of the entire equation to zero, we obtain:
\[2x - 2y - 2x\frac{dy}{dx} = 0\]
Dividing by 2 and rearranging to solve for \(\frac{dy}{dx}\):
\[x - y - x\frac{dy}{dx} = 0 \implies x\frac{dy}{dx} = x - y \implies \frac{dy}{dx} = \frac{x - y}{x}\]
- Calculating the Slope
Now, we substitute the coordinates of the point \((3, -1)\) into our derivative expression to find the slope \(m\) of the tangent line:
\[m = \left. \frac{dy}{dx} \right|_{(3, -1)} = \frac{3 - (-1)}{3} = \frac{4}{3}\]
Thus, the slope of the tangent line at \((3, -1)\) is \(m = \frac{4}{3}\).
- Determining the Tangent Line Equation
Using the point-slope form \(y - y_0 = m(x - x_0)\) with the point \((3, -1)\) and the slope \(m = \frac{4}{3}\), we have:
\[y - (-1) = \frac{4}{3}(x - 3)\]
\[y + 1 = \frac{4}{3}x - 4\]
Subtracting 1 from both sides, we get:
\[y = \frac{4}{3}x - 5\]
The equation of the tangent line is \(\boxed{y = \frac{4}{3}x - 5}\).
Examples
In engineering, understanding the slope of a curve at a specific point is fundamental for designing smooth transitions between different components, such as in road construction or mechanical part manufacturing. By finding the tangent line equation, designers can determine the exact orientation of a surface at a contact point, ensuring that structures align perfectly and mechanical stress is distributed correctly across joints. This application of calculus allows for precision in real-world engineering projects, ensuring stability and safety.