Final answer:
To find the points on the graph of the equation where the tangent line is horizontal, we need to find the derivative of the equation and set it equal to 0. The points where the tangent line is horizontal are (√48, 2√48) and (-√48, -2√48).
Step-by-step explanation:
To find the point(s) on the graph of the equation where the tangent line is horizontal, we need to find the derivative of the equation and set it equal to 0.
d/dx(x^2 - xy + y^2) = d/dx(48)
dy/dx(2y - x) = y - 2x
dy/dx = (y - 2x)/(2y - x)
To find the points where the tangent line is horizontal, we set dy/dx to 0:
(y - 2x)/(2y - x) = 0
y - 2x = 0
y = 2x
Substituting this equation back into the initial equation, we get:
x^2 - x(2x) + (2x)^2 = 48
-3x^2 + 4x^2 = 48
x^2 = 48
x = ±√48
Therefore, the points on the graph of the equation where the tangent line is horizontal are (√48, 2√48) and (-√48, -2√48).