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Manipal MET2012MathematicsApplication of Derivatives

Let P be any point on the curve x^ 2 / 3 +y^ 2 / 3 =a^ 2 / 3 . Then, the length of the segment of the tangent between the coordinate axes of length

Options

  1. A3 a
  2. B4 a
  3. C5 a
  4. Da

Correct answer

D. a

Step-by-step solution

Let the coordinates of the point P be (x₁, y₁ ) . This point lies on the curve aligned & x^ 2 / 3 +y^ 2 / 3 & =a^ 2 / 3 ....(i) & x₁^ 2 / 3 +y₁^ 2 / 3 & =a^ 2 / 3 ....(ii) aligned On differentiating Eq. (i) w.r.t. x , we get array rlrl 2 3 x^ -1 / 3 + 2 3 y^ -1 / 3 d y d x & =0 d y d x =- y^ 1 / 3 x^ 1 / 3 ( d y d x )_ (x₁, y₁ ) =- ( y₁ x₁ )^ 1 / 3 array The equation of the tangent at (x₁, y₁ ) to the given curve is aligned y-y₁ & =- y₁^ 1 / 3 x₁^ 1 / 3 (x-x₁ ) x x₁^ -1 / 3 +y y₁^ -1 / 3 & =x₁^ 2 / 3 +y₁^ 2 / 3 x x

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