Maths · Applications of derivatives: Rate of change of quantities, monotonic-Increasing and decreasing functions, Maxima and minima of functions of one variable
The two curves and
The two curves \( x^{3}-3 x y^{2}+2=0 \) and \( \mathbf{3} \boldsymbol{x}^{2} \boldsymbol{y}-\boldsymbol{y}^{3}-\boldsymbol{2}=\mathbf{0} \)
- A. cut at right angles
- B. touch each other
- C. cut at an angle \( \frac{\pi}{3} \)
- D. cut at an angle \( \frac{\pi}{4} \)
Step-by-step solution
Compute gradients: ∇f = (3(x^2-y^2), -6xy), ∇g = (6xy, 3(x^2-y^2)). Their dot product is 0, so normals are perpendicular, implying tangents are perpendicular. Hence curves cut orthogonally.
Related MCQs
- If the ratio of base radius and height of a cone is 1: 2 and percentage error in radius is then the error in its volume is…
- Function =(\boldsymbol{x}+\mathbf{2}) \boldsymbol{e}^{-\boldsymbol{x}} \) is…
- Equation of normal drawn to the graph of the function defined as =\frac{\sin x^{2}}{x} \) and =\mathbf{0} \) at the origin is?…
- The point on the curve which is nearest to (3,0) is…
- If the radius of a sphere is measured as with an error of then, find the approximate error in calculating its volume…