(a) Find the equation of the tangent to curve \(\frac{x^{2}}{4} + y^{2} = 1\) at point \(1, \frac{\sqrt{3}}{2}\). (b) Express \(\frac{3x + 2}{x^{2} + x - 2}\) in partial fractions.
Explanation
(a) \(\frac{x^{2}}{4} + y^{2} = 1\) Differentiating w.r.t x, \(\frac{\mathrm d y}{\mathrm d x} = -\frac{2x}{4(2y)} = -\frac{x}{4y}\) The gradient of the curve is \(\frac{\mathrm d y}{\mathrm d x} = -\frac{1}{4(\frac{\sqrt{3}}{2})}\) = \(-\frac{1}{2\sqrt{3}}\) Equation of the tangent at \((1, \frac{\sqrt{3}}{2})\) is \(\frac{y - \frac{\sqrt{3}}{2}}{x - 1} = -\frac{1}{2\sqrt{3}}\) \(1 - x = 2\sqrt{3} (y - \frac{\sqrt{3}}{2})\) \(1 - x = 2\sqrt{3} y - 3 \implies 2\sqrt{3} y + x - 4 = 0\). (b) \(\frac{3x + 2}{x^{2} + x - 2}\) \(x^{2} + x - 2 = x^{2} - x + 2x - 2\) \(x(x - 1) + 2(x - 1) \implies x^{2} + x - 2 \equiv (x - 1)(x + 2)\) \(\frac{3x + 2}{x^{2} + x - 2} = \frac{A}{x - 1} + \frac{B}{x + 2}\) = \(\frac{A(x + 2) + B(x - 1)}{(x - 1)(x + 2)}\) Comparing with the equation given, we have \(3x + 2 = A(x + 2) + B(x - 1)\) = \(Ax + 2A + Bx - B\) \(\implies A + B = 3 ... (1)\) \(2A - B = 2 ... (2)\) (1) + (2) : \(3A = 5 \implies A = \frac{5}{3}\) \(\frac{5}{3} + B = 3 \implies B = 3 - \frac{5}{3} = \frac{4}{3}\) \(\therefore \frac{3x + 2}{x^{2} + x - 2} = \frac{5}{3(x - 1)} + \frac{4}{3(x + 2)}\).