Question

Let \(P\) be any point on the curve \(x^{2/3}+y^{2/3}=a^{2/3}\). Then, what would be the length of the segment of the tangent between the coordinate axes?

• A. \(a\)
• B. \(2a\)
• C. \(3a\)
• D. \(4a\)
• E. \(5a\)

Hint:

A fundamental property of the astroid \(x^{2/3}+y^{2/3}=a^{2/3}\) is that the length of the tangent intercepted between the coordinate axes is always constant and equal to \(a\).

Answer 1

The Correct Option is A

Concept: The curve \[ x^{2/3}+y^{2/3}=a^{2/3} \] represents an astroid. A convenient parametric representation of this curve is \[ x=a\cos^3\theta, \qquad y=a\sin^3\theta \] The equation of the tangent at any point \(\theta\) is given by: \[ \frac{x}{a\cos\theta} + \frac{y}{a\sin\theta} = 1 \]

Step 1: Find the intercepts of the tangent. For the \(x\)-intercept (point \(A\)), set \(y=0\): \[ \frac{x}{a\cos\theta}=1 \implies x=a\cos\theta \] For the \(y\)-intercept (point \(B\)), set \(x=0\): \[ \frac{y}{a\sin\theta}=1 \implies y=a\sin\theta \]

Step 2: Find the length of the intercept between the axes. The length of the segment \(AB\) is calculated using the distance formula: \[ L = \sqrt{(a\cos\theta - 0)^2 + (0 - a\sin\theta)^2} \] \[ L = \sqrt{a^2\cos^2\theta + a^2\sin^2\theta} \] \[ L = a\sqrt{\cos^2\theta + \sin^2\theta} = a \]

Step 3: State the final result. The length of the segment of the tangent intercepted between the coordinate axes is constant and equal to \(a\). \[ \boxed{a} \]