Question

The portion of the tangent to the curve x\(^{\frac{2}{3}}\)+y\(^{\frac{2}{3}}\)=a\(^{\frac{2}{3}}\), a>0 at any point of it, intercepted between the axes

• A. varies at abscissa
• B. varies as ordinate
• C. is constant
• D. varies as the product of abscissa and ordinate

Answer 1

The Correct Option is C

Step 1: Parametrize the curve. 

The equation of the curve is \( x^{2/3} + y^{2/3} = a^{2/3} \). We can parametrize it using \( x = a\cos^3\theta \) and \( y = a\sin^3\theta \).

Step 2: Find dy/dx.

First, find \( \frac{dx}{d\theta} \) and \( \frac{dy}{d\theta} \):

\[ \frac{dx}{d\theta} = -3a\cos^2\theta\sin\theta \]

\[ \frac{dy}{d\theta} = 3a\sin^2\theta\cos\theta \]

Now, find \( \frac{dy}{dx} \):

\[ \frac{dy}{dx} = \frac{dy/d\theta}{dx/d\theta} = \frac{3a\sin^2\theta\cos\theta}{-3a\cos^2\theta\sin\theta} = -\frac{\sin\theta}{\cos\theta} = -\tan\theta \]

Step 3: Find the equation of the tangent line.

The equation of the tangent line at the point \( (a\cos^3\theta, a\sin^3\theta) \) is:

\[ y - a\sin^3\theta = -\tan\theta (x - a\cos^3\theta) \]

\[ y - a\sin^3\theta = -\frac{\sin\theta}{\cos\theta} (x - a\cos^3\theta) \]

\[ y\cos\theta - a\sin^3\theta\cos\theta = -x\sin\theta + a\cos^3\theta\sin\theta \]

\[ x\sin\theta + y\cos\theta = a\sin^3\theta\cos\theta + a\cos^3\theta\sin\theta \]

\[ x\sin\theta + y\cos\theta = a\sin\theta\cos\theta(\sin^2\theta + \cos^2\theta) \]

\[ x\sin\theta + y\cos\theta = a\sin\theta\cos\theta \]

Step 4: Find the x and y intercepts.

To find the x-intercept, set \( y = 0 \):

\[ x\sin\theta = a\sin\theta\cos\theta \]

\[ x = a\cos\theta \]

To find the y-intercept, set \( x = 0 \):

\[ y\cos\theta = a\sin\theta\cos\theta \]

\[ y = a\sin\theta \]

Step 5: Calculate the length of the intercept.

The length of the tangent intercepted between the axes is the distance between the points \( (a\cos\theta, 0) \) and \( (0, a\sin\theta) \), which is:

\[ L = \sqrt{(a\cos\theta - 0)^2 + (0 - a\sin\theta)^2} = \sqrt{a^2\cos^2\theta + a^2\sin^2\theta} = \sqrt{a^2(\cos^2\theta + \sin^2\theta)} = \sqrt{a^2} = a \]

Final Answer

Since \( L = a \), the length of the tangent intercepted between the axes is constant.