Question

Find the points on the ellipse 5x2 + y2 = 5 that are farthest away from the point (1, 0).

smaller y-value (x,y) = ___________
larger y-value (x,y) = __________

Answer 1

To find the points on the ellipse 5x² + y² = 5 that are farthest away from the point (1, 0), we can take the derivative of the equation and solve for x. The points (0, √(5)) and (0, -√(5)) have the largest and smallest y-values on the ellipse, respectively.

Step-by-step explanation:

To find the points on the ellipse 5x² + y² = 5 that are farthest away from the point (1, 0), we can start by expressing the equation in terms of y. Rearranging the equation, we get y = √(5 - 5x²), which represents the upper half of the ellipse.

The farthest points away from the point (1, 0) will be the points on the ellipse with the largest y-values. To find these points, we can take the derivative of y with respect to x, set it equal to zero, and solve for x.

Taking the derivative of y with respect to x, we get dy/dx = (-10x)/√(5 - 5x²). Setting this equal to zero, we have -10x = 0. Solving for x, we get x = 0.

Substituting x = 0 into the equation y = √(5 - 5x²), we get y = √5.

Therefore, the point (0, √5) is on the ellipse and has the largest y-value. The point (-0, -√5) is also on the ellipse and has the smallest y-value.

Answer 2

To find the points on the ellipse 5x^2 + y^2 = 5 that are farthest away from the point (1, 0), we can use the concept of distance formula and calculus. The farthest points on the ellipse from (1, 0) occur at the endpoints of the major axis. So, one point with the smaller y-value is (-sqrt(5), 0), and the point with the larger y-value is (sqrt(5), 0).

Step-by-step explanation:

To find the points on the ellipse 5x^2 + y^2 = 5 that are farthest away from the point (1, 0), we can use the concept of distance formula and calculus.

We start by rearranging the equation of the ellipse to the standard form: x^2/1 + y^2/5 = 1. Therefore, the center of the ellipse is at (0, 0) and the semi-major axis is sqrt(5) and the semi-minor axis is 1.

The farthest points on the ellipse from (1, 0) occur at the endpoints of the major axis. So, one point with the smaller y-value is (-sqrt(5), 0), and the point with the larger y-value is (sqrt(5), 0).