100 POINTS! Using the following equation, find the center and radius of the circle. You must show and explain all work and calcu

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2 years ago

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2 answers:

omeli [17] · Answer 1 · 2023-07-02T12:38:13+03:00

Answer:

Center: (-4, 1)

Radius: √2

Explanation:

Circle general equation is represented by (x - h)² + (y - k)² = r²

where (h, k) is the center and "r" resembles radius

To write in general equation form, start completing square:

x² + y² + 8x -2y + 15 = 0

x² + 8x + y² - 2y + 15 = 0

(x + 4)² - 4² + (y - 1)² - (-1)² + 15 = 0

(x + 4)² + (y - 1)² - 16 - 1 + 15 = 0

(x + 4)² + (y - 1)² - 2 = 0

General equation form:

(x - (-4))² + (y - 1)² = (√2)²

From here, comparing with the formula equation.

It is determined: center: (-4, 1) and radius: √2

galina1969 [7] · Answer 2 · 2023-07-02T14:42:29+03:00

Answer:

center = (-4, 1)

radius = √2

Step-by-step explanation:

Equation of a circle

$(x-a)^2+(y-b)^2=r^2$

where:

Given equation:

$x^2+y^2+8x-2y+15=0$

Subtract 15 from both sides to move the constant to the right side of the equation:

$\implies x^2+y^2+8x-2y=-15$

Rearrange the left side to collect like variables:

$\implies x^2+8x+y^2-2y=-15$

Add the square of half the coefficients of the variables x and y to both sides:

$\implies x^2+8x+\left(\dfrac{8}{2}\right)^2+y^2-2y+\left(\dfrac{-2}{2}\right)^2=-15+\left(\dfrac{8}{2}\right)^2+\left(\dfrac{-2}{2}\right)^2$

$\implies x^2+8x+16+y^2-2y+1=2$

Factor the two perfect square trinomials on the left side:

$\implies (x+4)^2+(y-1)^2=2$

Compare the found equation with the general form to find the center and radius of the circle:

Learn more about circle equations here: