Find the shortest distance between the curves y^2=x^3 and 9 x^2+9 y^2-30 y+16=0 .

Name: Problem 45, Chapter 8: Differential Calculus Booster with Problems and Solutions for JEE Main and Advanced
Uploaded: 2022-02-28T12:00:37-08:00
Duration: 3 min 22 s
Channel: Varsha Aggarwal
Description: Find the shortest distance between the curves y^2=x^3 and 9 x^2+9 y^2-30 y+16=0 .

step 1 So given curve are y square equal x cube and 9 x squared plus 9 y square minus 30 y plus 16 equa

Find the shortest distance between the curves y^2=x^3 and 9...

Key Concepts

Geometric Analysis of Curves

Geometric analysis involves studying the inherent properties of curves, such as symmetry, curvature, and specific features characteristic of conic sections or algebraic curves. Understanding these properties can provide insights into simplifying the problem, revealing natural coordinate transformations or symmetries that reduce the complexity of locating the shortest distance between the curves.

Implicit Differentiation and Tangency Conditions

Implicit differentiation is a method used to find the derivative of a function defined by an implicit equation, where the dependent variable is not isolated. This technique is crucial in analyzing the slopes of curves, and in the context of minimizing distance, the condition that the line connecting points on two curves is perpendicular to the tangents (or has a specific slope relationship) often leads to the necessary equations that reveal the optimal points.

Constrained Optimization

Constrained optimization is used when the variables involved in an optimization problem are subject to specific conditions or restrictions, such as being confined to lie on given curves. In such cases, techniques like substitution or Lagrange multipliers are applied to incorporate the constraints into the optimization process, ensuring that the solutions are valid within the prescribed conditions.

Distance Formula and Function Minimization

The distance formula calculates the distance between two points in a plane, and when applied to points on different curves, it creates a function whose minimum value represents the shortest distance between the curves. Minimizing this function often involves working with the square of the distance to avoid square roots, which simplifies differentiation and subsequent optimization steps.

Curve Parametrization

Curve parametrization involves expressing the coordinates of points on a curve using one or more parameters. This approach simplifies complex curves by reducing their description to functions of a single variable, making it easier to analyze properties like distance or slope and to carry out subsequent calculus operations such as differentiation and integration.

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