Parabola
Common Tangent
Grade 11
Question:
<p>The equation of the common tangent to \(y^2 = 4ax\) and \(x^2 = 4ay\), where \(a > 0\), is:</p>
<p>(a) \(x + y + a = 0\)</p>
<p>(b) \(x + y - a = 0\)</p>
<p>(c) \(x - y + a = 0\)</p>
<p>(d) \(x - y - a = 0\)</p>
Step-by-Step Solution
Key Concept: A common tangent to two parabolas must satisfy the tangent conditions for both curves simultaneously.
<p>For the parabola \(y^2 = 4ax\), a tangent with slope \(m\) is \(y = mx + \frac{a}{m}\). For the parabola \(x^2 = 4ay\), a tangent can be written as \(x = my + a/m\). For a common tangent to both parabolas, we solve these simultaneously. The common tangent is \(x + y - a = 0\).</p>
Correct Answer: b