A circle is defined by x^2 + y^2 + 2x-4y. A line defined by 5x+9y=66, intersects the circle at E & F?

Question

Find the length of the chord EF

Answer 1

I think you mean the equation of the circle is:
x² + y² + 2x - 4y = 0
(x² + 2x) + (y² - 4y) = 0
(x² + 2x + 1) + (y² - 4y + 4) = 1 + 4
(x + 1)² + (y - 2)² = 5

The center of the circle is (-1, 2) and radius of the circle is √5.
_______

The equation of the line is:
5x + 9y - 66 = 0

The distance from the center of the circle to the line is:

d = | 5*(-1) + 9*2 - 66 | / √(5² + 9²)
d = | -5 + 18 - 66 | / √(25 + 81)
d = 53 / √106 ≈ 5.15

The closest point of the line to the center of the circle is greater than the the radius. Consequently the line and circle do not intersect.