Question

The shortest distance between the line \(y = x\) and the curve \(y^2 = x - 2\) is :

• A. \(\frac{7}{4\sqrt{2}}\)
• B. \(\frac{7}{8}\)
• C. \(\frac{11}{4\sqrt{2}}\)
• D. \(2\)

Hint:

Take a general point \((t^2 + 2, t)\) on the parabola and minimise its distance to \(x - y = 0\).

Answer 1

The Correct Option is A

A point on the curve: \((t^2 + 2,\ t)\). Its distance from \(x - y = 0\) is \(d = \frac{|t^2 + 2 - t|}{\sqrt{2}} = \frac{t^2 - t + 2}{\sqrt{2}}\) (the quadratic is always positive).
\(t^2 - t + 2 = \left(t - \frac{1}{2}\right)^2 + \frac{7}{4}\), with minimum value \(\frac{7}{4}\) at \(t = \frac{1}{2}\).
\(d_{min} = \frac{7}{4\sqrt{2}}\).
Answer: (1) \(\frac{7}{4\sqrt{2}}\)