Equation of the tangent to the hyperbola at $\left(-1, -\frac{1}{2}\right)$ is
Step-by-Step Solution
Key Concept: To find the parameter in a conic equation, substitute the known point; then use implicit differentiation or the tangent formula at the desired point.
The hyperbola $2x^2 + 2y^2 + 5xy + \lambda = 0$ passes through $(1,1)$, so $2 + 2 + 5 + \lambda = 0$, giving $\lambda = -9$. The hyperbola is $2x^2 + 2y^2 + 5xy = 9$. The tangent at $(-1, \frac{7}{2})$ is found using $2x(-1) + 2y(\frac{7}{2}) + 5(\frac{x+(-1)y}{2}) = 9$, which simplifies to $3x + 2y = 4$.
Correct Answer: 2