The equation ..x ^2 − 8x + y^ 2 + 4y + 11 = 0....can any1 help me plzz ??

Question

The equation

x ^2 − 8x + y ^2 + 4y + 11 = 0

represents a circle.

a) Find its centre and radius.

(b) Find the coordinates of any points at which the circle in part (a) intersects the line y = −x − 1.

Answer 1

For part a, you need to put the equation in standard form:
x² - 8x (leave a gap) + y² + 4y (leave a gap) = -11
to fill in the gaps, you complete the square
divide coefficient of x by 2 and then square it.
First gap: (8/2)² = 16
Second gap (4/2)² = 4
You have to add these numbers to both sides.
x² - 8x + 16 + y² + 4y + 4 = -11 + 16 + 4
Now factor the expressions on the left and add the numbers on the right.
(x - 4)² + (y + 2)² = 9

The center is (4, -2) and the radius is SQRT(9) = 3.

To find points of intersection (part b)
Since y = -x - 1, you should replace all the y variables in the original equation with -x - 1. This gives us:
x² - 8x + (-x - 1)² + 4(-x - 1) + 11 = 0
x² - 8x + x² + 2x + 1 - 4x - 4 + 11 = 0 (Simplify parentheses)
2x² -10x + 8 = 0 (add like terms)
x² - 5x + 4 = 0 (divide all terms by 2)
(x - 1)(x - 4) = 0 (factor)
x = 1, 4.

These are the values of x at the points of intersection.
to find y, use y = -x - 1
When x = 1, y = -1 - 1 = -2
When x = 4, y = -4 - 1 = -5
So the points of intesection are (1, -2) and (4, -5)

Answer 2

Standard form of a circle is
(x - h)² + (y - k)² = r²
where (h, k) is the center and r is the radius

x² − 8x + y² + 4y + 11 = 0
x² − 8x + y² + 4y = -11
x² − 8x + 16 + y² + 4y + 4 = -11 + 16 + 4
(x - 4)² + (y + 2)² = 9

Center is (4, -2) radius is 3

Find point(s) of intersection of
x² − 8x + y² + 4y + 11 = 0
y = −x − 1

x² − 8x + (-x - 1)² + 4(-x - 1) + 11 = 0
x² − 8x + x² + 2x + 1 - 4x - 4 + 11 = 0
2x² − 10x + 8 = 0
2(x² − 5x + 4) = 0
2(x - 4)(x - 1) = 0
x = 4 or x = 1

y = −x − 1
y = −4 − 1 or y = −1 − 1
y = -5 or -2

Points of intersection are (4, -5) and (1, -2)

Answer 3

x + y = 2 x = 2 - y 4y^2 - (-y+2)^2 = eleven 4y^2 - (y^2 - 4y + 4) = eleven 3y^2 + 4y - 4 = eleven 3y^2 + 4y - 15 = 0 (3y - 5)(y + 3) = 0 3y - 5 = 0 3y = 5 y = 5/3 y + 3 = 0 y = -3 x + y = 2 x + -3 = 2 x = 5 x + y = 2 x + 5/3 = 6/3 x = a million/3 the achieveable ideas are (5, -3) and (a million/3, 5/3).

Answer 4

x^2 - 8x + 7 + y^2 + 4y + 4 = 0

x^2 - 8x + 16 + y^2 + 4y + 4 = 9

(x-4)^2 + (y+2)^2 = 3^2

radius = 3

center = (4,-2)

Answer 5

I'll just tell you one way you can solve the problem. You'll have to do the work yourself....

a) complete the square and factor x and y. That will get you to the standard circle formula. From there, you can easily figure out what the center is, and the radius is.

b) plug the line formula y= into the circle formula and solve for x. Once you have x, solve for y using those x.

Answer 6

C = 4.82
R = 3.93
possible coords. = (6, 13)

Answer 7

9y

Answer 8

sorry no , i am just starting algabra lol , ( joke)