Verify that points C(-2,3) and D(2 √(2), √(6)) are points on the ellipse with foci at A(-2,0) an

step 1 In this question, the given points are point A, which is minus 2 .0, point B, which is 2 .0, poi

Verify that points C(-2,3) and D(2 √(2), √(6)) are points on...

Verify that points $C(-2,3)$ and $D(2 \sqrt{2}, \sqrt{6})$ are points on the ellipse with foci at $A(-2,0)$ and $B(2,0),$ by verifying $d(A C)+d(B C)=d(A D)+d(B D) .$ The expression that results has the form $\sqrt{A+B}+\sqrt{A-B},$ which prior to the common use of technology had to be simplified using the formula $\sqrt{A+B}+\sqrt{A-B}=\sqrt{a+\sqrt{b}}$ where $a=2 A$ and $b=4\left(A^{2}-B^{2}\right) .$ Use this relationship to verify the equation above.

Key Concepts

Ellipse Definition

An ellipse is defined as the locus of points for which the sum of the distances from any point on the ellipse to two fixed points (called the foci) is constant. This definition is foundational in understanding and verifying whether a given point lies on a specific ellipse by checking that the distance sum property holds true.

Distance Formula

The distance formula, derived from the Pythagorean theorem, is used to calculate the Euclidean distance between two points in the coordinate plane. This formula is essential for computing the distances from any point on the ellipse to the foci, enabling the verification of the ellipse's constant sum property.

Radical Expression Simplification

When dealing with expressions involving square roots, such as ?(A+B) + ?(A?B), simplifying these expressions into a single radical form can be highly beneficial to reduce complexity. The provided formula, ?(A+B) + ?(A?B) = ?(2A + ?(4A² ? 4B²)), is an example of algebraic manipulation used to combine and simplify radical expressions, an important technique in pre-digital computation.

Verification of the Constant Sum Property

To confirm that a given point lies on an ellipse, one must calculate the sum of the distances from that point to each focus and verify that it equals the fixed constant of the ellipse. This verification process shows that the point satisfies the fundamental defining property of the ellipse, thereby establishing its membership on the conic.

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