Q95 / 155 · WBJEE 2023 · Mathematics

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 as abscissa
  • B
    varies as ordinate
  • Correct answer
    C
    is constant
  • D
    varies as the product of abscissa and ordinate

Solution

Hint : for the given curve \(x^{\frac{2}{3}}+y^{\frac{2}{3}}=a^{\frac{2}{3}}, a > 0\) Parametric Coordinates are \(x=a \cos ^3 \theta\) \(y=a \sin ^3 \theta\) \(\begin{gathered} \frac{d y}{d x}=\text { slope of tangent at any }=\frac{d y / d \theta}{d x / d \theta} \\ \text { point }(x, y) \end{gathered}\) \(=\frac{3 a \sin ^2 \theta \cdot \cos \theta}{3 a \cos ^2 \theta(-\sin \theta)}=-\tan \theta\) Equation of tangent at \(\left(a \cos ^3 \theta, a \sin ^3 \theta\right)\) \(y-a \sin ^3 \theta=-\tan \theta\left(x-a \cos^3 \theta\right)\) its \(x\)-intercept \(=a \cos \theta\) its \(y\)-intercept \(=a \sin \theta\) So, the tangent cuts the axes at \(A(a \cos \theta, 0)\) and \(B(0, a \sin \theta)\) respectively. \(A B=\sqrt{(a \cos \theta)^2+(a \sin \theta)^2}=a\), which is constant

Hence, the correct answer is option: 'C'.

Relevant Questions