Parabola
Parabola with Given Focus and Directrix
Grade 11
Question:
<p>The equation of the Parabola P whose directrix is the common tangent to \(y^2 = 4ax\) and \(x^2 = 4ay\), and whose focus is the point which divides OA internally in the ratio \((1 + \sqrt{3}):(7 - \sqrt{3})\), is:</p>
<p>(a) \((x - y)^2 = (2 + \sqrt{3})a(x + y - (1 + \sqrt{3})a)\)</p>
<p>(b) \((x - y)^2 = (2 + \sqrt{3})a(2x + 2y - (2 + \sqrt{3})a)\)</p>
<p>(c) \((x - y)^2 = (2 + \sqrt{3})a(2x + 2y - (1 + \sqrt{3})a)\)</p>
<p>(d) \((x - y)^2 = (2 - \sqrt{3})a(x + y - (1 + \sqrt{3})a)\)</p>
Step-by-Step Solution
Key Concept: Apply the fundamental definition of a parabola: every point on it is equidistant from the focus and directrix.
<p>Using the definition of a parabola (locus of points equidistant from focus and directrix), with directrix \(x + y - a = 0\) and focus at the point dividing OA in ratio \((1 + \sqrt{3}):(7 - \sqrt{3})\), the equation of parabola P is derived.</p>
Correct Answer: c