From Exercise 30, verify the points (4, (10)/(3)) and (-3,-√(15)) are on the ellipse defined by

step 1 In this question, the equation given is 5x square plus 9 by square equals to 180 and the points

From Exercise 30, verify the points (4, (10)/(3)) and...

From Exercise $30,$ verify the points $\left(4, \frac{10}{3}\right)$ and $(-3,-\sqrt{15})$ are on the ellipse defined by $5 x^{2}+9 y^{2}=180$ Then verify that $d_{1}+d_{2}=d_{3}+d_{4}$. (GRAPH CAN'T COPY)

Key Concepts

Ellipse and Its Standard Equation

An ellipse is a conic section defined as the set of all points in the plane for which the sum of the distances from two fixed points (the foci) is constant. Its standard equation is derived by rewriting the quadratic form in a way that highlights its geometric properties, often involving a denominator under x² and y². Recognizing the ellipse equation is fundamental in coordinate geometry and helps in understanding the shape, orientation, and key features such as the center, major and minor axes, and foci.

Verification of Points by Substitution

In analytic geometry, verifying whether a point lies on a given curve involves substituting the coordinates of the point into the curve's equation. If the equation holds true after the substitution, then the point is on the curve. This method is a straightforward application of algebra and is used across various contexts to confirm the validity of solutions and to check if specific points satisfy the defining equation of the curve.

Distance Formula

The distance formula, derived from the Pythagorean theorem, is used to calculate the distance between two points in a coordinate plane. It is given by the square root of the sum of the squares of the differences in the x-coordinates and y-coordinates of the two points. This formula is essential not only in verifying geometric properties like those of an ellipse but also in many other areas of mathematics where measuring exact distances between points is required.

Constant Sum Property of an Ellipse (Focal Property)

A defining characteristic of an ellipse is that for any point on the ellipse, the sum of the distances to the two foci is constant. This property is central to the ellipse's definition and is often used to derive other properties of the ellipse. In problems involving distances from the foci, verifying that the sum remains constant serves as a confirmation of the ellipse's geometric integrity and provides a practical application of the distance formula.

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