The point on the curve 3 x^2-4 y^2=72 which is nearest to the line 3 x+2 y+1=0 is (A) (6,-3) (B)

step 1 In this question we have to find point on curves x -skar minus 4 y -square equals 72, which is n

The point on the curve 3 x^2-4 y^2=72 which is nearest to...

Key Concepts

Conic Sections

Conic sections—curves like circles, ellipses, parabolas, and hyperbolas—are obtained by intersecting a plane with a cone. In problems like this, the given curve is a conic, and its geometric properties are used in combination with optimization techniques to find relevant points, such as the closest point to a given line.

Lagrange Multipliers

Lagrange multipliers is a strategy used in constrained optimization to find local maxima and minima of a function subject to equality constraints. By introducing an auxiliary variable (or multiplier), one can transform the constrained problem into an unconstrained one by setting the gradient of the objective function equal to the gradient of the constraint multiplied by the Lagrange multiplier.

Constrained Optimization

This is the process of finding an optimum (minimum or maximum) of an objective function subject to certain restrictions or constraints. In this context, it involves minimizing the distance from a point to a line while ensuring that the point remains on a specified curve.

Distance from a Point to a Line

This concept involves calculating the shortest distance between a given point and a line, typically using the formula |Ax + By + C| / sqrt(A^2 + B^2) for a line described by Ax + By + C = 0. It serves as the primary measure to optimize when finding the nearest point on a curve to a given line.

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