Final answer:
To determine which point lies in the solution set, substitute the coordinates of each point into the equation and check if the inequality is satisfied. The point that satisfies the inequality is b) (3, -2.5).
Step-by-step explanation:
To determine which point lies in the solution set of the equation (x−2)²/25 + (y+3)²/4 < 1, we can substitute the coordinates of each point into the equation and check if the inequality is satisfied.
a) (x, y) = (4, -0.5)
Substituting into the equation, we get (4-2)²/25 + (-0.5+3)²/4 = 4/25 + (2.5)²/4 = 0.16 + 1.56 = 1.72 > 1. Therefore, point a) does not lie in the solution set.
b) (x, y) = (3, -2.5)
Substituting into the equation, we get (3-2)²/25 + (-2.5+3)²/4 = 1/25 + (0.5)²/4 = 0.04 + 0.0625 = 0.1025 < 1. Therefore, point b) lies in the solution set.
c) (x, y) = (-2.5, 4)
Substituting into the equation, we get (-2.5-2)²/25 + (4+3)²/4 = (-4.5)²/25 + 7²/4 = 20.25/25 + 49/4 = 0.81 + 12.25 = 13.06 > 1. Therefore, point c) does not lie in the solution set.
d) (x, y) = (-4.5, -3)
Substituting into the equation, we get (-4.5-2)²/25 + (-3+3)²/4 = (-6.5)²/25 + 0²/4 = 42.25/25 + 0 = 1.69 > 1. Therefore, point d) does not lie in the solution set.
Therefore, the point that lies in the solution set is b) (3, -2.5).