Problem

Problem

The common tangent to the parabola y2=4axy^2 = 4ax and x2=4ayx^2 = 4ay is a. x+y+a=0x + y + a = 0 b. x+y−a=0x + y - a = 0 c. x−y+a=0x - y + a = 0 d. x−y−a=0x - y - a = 0
Intuitive explanation

The key is recognizing that a common tangent touches both curves simultaneously, so it must satisfy the tangent condition for each parabola at potentially different points. Write the general tangent line to each parabola, then use the constraint that they're the same line to find the slope and intercept. This reduces to solving for which line geometry allows both parabolas to "kiss" it at once.

Intuitive explanation

The key is recognizing that a common tangent touches both curves simultaneously, so it must satisfy the tangent condition for each parabola at potentially different points. Write the general tangent line to each parabola, then use the constraint that they're the same line to find the slope and intercept. This reduces to solving for which line geometry allows both parabolas to "kiss" it at once.

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