From Exercise 29, verify the points (-3,2) and (√(12), 0) are on the ellipse defined by 4 x^2+3

step 1 In this question, the given equation is 4x squared plus 3x squared equals to 48 and 2 points giv

From Exercise 29, verify the points (-3,2) and (√(12), 0)...

Key Concepts

Distance Formula in Coordinate Geometry

The distance formula is used to compute the distance between two points in the coordinate plane. By applying the formula, one can calculate distances (d1, d2, etc.) between key points, such as foci, vertices, or other points on the ellipse, which is essential in establishing relationships like the constant sum property for ellipses.

Ellipse Focal Property

A defining property of ellipses is that for any point on the curve, the sum of its distances to the two foci is constant. This principle underpins many geometric proofs and verifications involving ellipses and is used to show equalities between sums of distances such as d1 + d2 and d3 + d4, highlighting the inherent symmetry and balance in elliptical geometry.

Ellipse Equation Standard Form

This concept involves rewriting the equation of an ellipse into its standard form so that its geometric properties can be clearly recognized. Converting an ellipse from a general quadratic form to standard form allows one to easily identify the lengths of the semi-major and semi?minor axes, the center, and the orientation, which are all crucial for analyzing points and distances related to the ellipse.

Verification of a Point on a Conic

Verifying whether a given point lies on a conic section, such as an ellipse, is done by substituting the point’s coordinates into the equation of the conic. If the equation is satisfied, the point is on the conic. This process is a fundamental method in analytic geometry for confirming membership of a point on any defined curve.

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