Hyperbola
Tangent Equation
Grade 11
Question:
<p>For the hyperbola \(H\) as described above, find the equation of the tangent to \(H\) at the point \(\left(-1, \frac{7}{2}\right)\) on it.</p>
<p>(a) \(3x + 2y = 2\)</p>
<p>(b) \(3x + 2y = 4\)</p>
<p>(c) \(4x + 2y = 9\)</p>
<p>(d) \(6x + 4y = 7\)</p>
Step-by-Step Solution
Key Concept: Apply the tangent formula to the hyperbola equation at the given point to obtain the tangent line equation.
<p><strong>Solution:</strong> Once the equation of the hyperbola \(H\) is determined from the previous conditions, the tangent at a point \((x_0, y_0)\) on the hyperbola is given by the tangent equation for that hyperbola. For the point \(\left(-1, \frac{7}{2}\right)\), substituting into the tangent formula derived from the hyperbola equation obtained in the previous part yields \(3x + 2y = 4\).</p><p>We can verify: \(3(-1) + 2\left(\frac{7}{2}\right) = -3 + 7 = 4\). ✓</p><p>∴ Answer is (b).</p>
Correct Answer: b