Applications of Derivatives
Distance Optimization
Grade 12
Question:
<p>Minimum distance between the curves <span class="math">\(y^2 = x - 1\)</span> and <span class="math">\(x^2 = y - 1\)</span> is equal to:</p>
<p>(a) <span class="math">\(\frac{2}{4}\)</span></p>
<p>(b) <span class="math">\(\frac{3\sqrt{2}}{4}\)</span></p>
<p>(c) <span class="math">\(\frac{5\sqrt{2}}{4}\)</span></p>
<p>(d) <span class="math">\(\frac{7\sqrt{2}}{4}\)</span></p>
Step-by-Step Solution
Key Concept: Recognize the symmetry of the two parabolas about the line y = x, and use distance minimization with calculus or parametric analysis.
<p>The curves are a parabola opening rightward and another parabola. By symmetry, observe that <span class="math">$y^2 = x - 1$</span> and <span class="math">$x^2 = y - 1$</span> are reflections about the line <span class="math">$y = x$</span>. The minimum distance occurs along the perpendicular direction. Use parametric forms and optimization to find the minimum distance is <span class="math">$\frac{3\sqrt{2}}{4}$</span>.</p>
Correct Answer: b