© All Rights reserved @ LearnWithDash
Step 1: Identify the focus and a point on the parabola. The given parabola is $ y^2 = 4a x $. The focus of this parabola is $ (a, 0) $. A point on the parabola can be represented as $ (x_1, y_1) $ where $ y_1^2 = 4a x_1 $.
Step 2: Find the midpoint of the line joining the focus and the point on the parabola. The midpoint $ M $ of the line joining $ (a, 0) $ and $ (x_1, y_1) $ is given by $ M = \left( \frac{a + x_1}{2}, \frac{y_1}{2} \right) $.
Step 3: Determine the locus of the midpoint. Since $ y_1^2 = 4a x_1 $, we have $ x_1 = \frac{y_1^2}{4a} $. Substitute $ x_1 $ in the expression for the midpoint: $ M = \left( \frac{a + \frac{y_1^2}{4a}}{2}, \frac{y_1}{2} \right) = \left( \frac{4a^2 + y_1^2}{8a}, \frac{y_1}{2} \right) $.
Step 4: Simplify the expression. Let $ h = \frac{4a^2 + y_1^2}{8a} $ and $ k = \frac{y_1}{2} $. Then $ y_1 = 2k $ and $ y_1^2 = 4k^2 $. Substitute into $ h $: $ h = \frac{4a^2 + 4k^2}{8a} = \frac{a^2 + k^2}{2a} $.
Step 5: Form the equation of the locus. Rearrange to get $ 2a h = a^2 + k^2 $, which simplifies to $ k^2 = 2a h - a^2 $. This is the equation of a parabola with the directrix $ x = 0 $.
