The conic represented by x=2(cost+sint),y=5(cost−sint) isa circlea parabolaan ellipsea hyperbola

The given parametric equations are x = 2(cos(t) + sin(t)) and y = 5(cos(t) - sin(t)).

To identify the conic section represented by these equations, we can eliminate the parameter t by using the Pythagorean identity sin^2(t) + cos^2(t) = 1.

First, we can express cos(t) and sin(t) in terms of x and y:

cos(t) = (x - y) / 7 and sin(t) = (x + y) / 7.

Then, we substitute these expressions into the Pythagorean identity:

[(x - y) / 7]^2 + [(x + y) / 7]^2 = 1.

Simplify this equation to get:

(x^2 - 2xy + y^2) / 49 + (x^2 + 2xy + y^2) / 49 = 1.

Combine like terms:

2x^2 + 2y^2 = 49.

Divide by 2:

x^2 + y^2 = 24.5.

This is the standard form of the equation of a circle with radius sqrt(24.5). So, the conic represented by the given parametric equations is a circle.