Ellipse
Directrix and standard form
Grade 11
Question:
<p>The eccentricity of an ellipse, with centre at the origin, is \(1/2\). If one directrix is \(x = 4\), the equation of the ellipse is</p>
<p>(a) \(3x^2 + 4y^2 = 1\)</p>
<p>(b) \(3x^2 + 4y^2 = 12\)</p>
<p>(c) \(4x^2 + 3y^2 = 1\)</p>
<p>(d) \(4x^2 + 3y^2 = 12\)</p>
Step-by-Step Solution
Key Concept: Use the relationship between directrix, semi-major axis, and eccentricity to find the equation.
<p>Given: \(e = 1/2\) and directrix \(x = 4\).</p><p>For an ellipse, directrix is at \(x = \frac{a^2}{c}\) where \(c = ae\).</p><p>So \(\frac{a^2}{ae} = 4\), which gives \(\frac{a}{e} = 4\), thus \(a = 4e = 2\).</p><p>Since \(e = 1/2\), we have \(c = ae = 2 \times 1/2 = 1\).</p><p>Thus \(b^2 = a^2 - c^2 = 4 - 1 = 3\).</p><p>The equation is \(\frac{x^2}{4} + \frac{y^2}{3} = 1\), or \(3x^2 + 4y^2 = 12\).</p>
Correct Answer: B