Determine the equation of the circle graphed below .

( help please )

(x + 5)² + (y - 4)² = 17

Step-by-step explanation:

The equation of a circle in standard form is

(x - h)² + (y - k)² = r²

where (h, k ) are the coordinates of the centre and r is the radius

Here (h, k ) = (- 5, 4 ) , then

(x - (- 5) )² + (y - 4)² = r² , that is

(x + 5)² + (y - 4)² = r²

r is the distance from the centre to a point on the circle

Calculate r using the distance formula

r =

with (x₁, y₁ ) = (- 5, 4) and (x₂, y₂ ) = (- 1, 5)

r =

  =

  =

  = ⇒ r² = ( )² = 17

Then

(x + 5)² + (y - 4)² = 17 ← equation of circle

x² + y² + 10x - 8y + 24 = 0

Step-by-step explanation:

equation of a circle :-

equation of a circle :-(x - h)² + (y - k)² = r²

for this circle :-

finding radius of the circle

we've been given a point ( -1, 5) that lies on the circle and hence should satisfy the equation of the circle

putting x = -1 and y = 5

=> (-1 - (-5))² + (5 - 4)² = r²

=> (-1 + 5)² + 1² = r²

=> 4² + 1 = r²

=> 17 = r²

the radius if the circle is √17 units

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finding the equation of the circle

=> (x + 5)² + (y - 4)² = (√17)²

=> x² + 25 + 10x + y² + 16 - 8y = 17

=> x² + y² + 10x - 8y + 24 = 0

(b) 240

step-by-step explanation:

first, we use the pythagorean formula using the given side lengths to find x. then we use x to find the side lengths. then we add the side lengths to find the perimeter.

pythagorean theorem formula:

a^2 + b^2 = c^2

(x - 20)^2 + (x - 40)^2 = x^2

x^2 - 40x + 400 + x^2 - 80x + 1600 = x^2

x^2 -120x + 2000 = 0

(x - 100)(x - 20) = 0

x - 100 = 0 or x - 20 = 0

x = 100 or x = 20

we see that the solution x = 20 must be discarded because it will give a negative side length and a side length of 0. the only valid solution is x = 100.

perimeter = sum of the lengths of the sides

perimeter = x + x - 20 + x - 40

perimeter = 3x - 60

replace x with 100.

perimeter = 3(100) - 60

perimeter = 300 - 60

perimeter = 240