A circle has equation
x2 +y2 + 4x - 2y - 11 = 0
the coordinates of the points where C crosses the y axis, giving your answers in simplified surds
Find the coordinates
these points are where x=0, right?
So just solve
y^2 - 2y - 11 = 0
To find the coordinates where the circle crosses the y-axis, we need to substitute x = 0 into the equation of the circle.
Let's substitute x = 0 into the given equation:
(0)^2 + y^2 + 4(0) - 2y - 11 = 0
y^2 - 2y - 11 = 0
Now, we can solve this quadratic equation to find the y-coordinates:
Using the quadratic formula, y = (-b ± √(b^2 - 4ac)) / 2a, where a = 1, b = -2, and c = -11.
y = (-(-2) ± √((-2)^2 - 4(1)(-11))) / 2(1)
y = (2 ± √(4 + 44)) / 2
y = (2 ± √48) / 2
y = (2 ± 4√3) / 2
y = 1 ± 2√3
Therefore, the coordinates where the circle crosses the y-axis are (0, 1 + 2√3) and (0, 1 - 2√3).
To find the coordinates where the circle crosses the y-axis, we need to determine the points where the x-coordinate is equal to zero. As the y-axis is the vertical line with an x-coordinate of zero, substituting x = 0 into the equation of the circle will give us the y-coordinates.
Let's start by substituting x = 0 into the equation of the circle:
0^2 + y^2 + 4(0) - 2y - 11 = 0
Simplifying the equation, we have:
y^2 - 2y - 11 = 0
Now, we need to solve this quadratic equation to find the values of y. We can do this by factoring or using the quadratic formula.
Using the quadratic formula, the general form is:
y = (-b ± √(b^2 - 4ac)) / (2a)
For our equation y^2 - 2y - 11 = 0, the coefficients are: a = 1, b = -2, and c = -11.
Substituting these values into the quadratic formula, we get:
y = (-(-2) ± √((-2)^2 - 4(1)(-11))) / (2(1))
Simplifying further, we have:
y = (2 ± √(4 + 44)) / 2
y = (2 ± √48) / 2
y = (2 ± √16 * 3) / 2
y = (2 ± 4√3) / 2
Now, we can simplify the expressions:
y = 2/2 ± 4√3/2
y = 1 ± 2√3
So, the coordinates of the points where the circle crosses the y-axis are (0, 1 + 2√3) and (0, 1 - 2√3).