Question

Determine the equation of the circle graphed below .

( help please )

Answer 1

(x + 5)² + (y - 4)² = 17

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 =
`\sqrt{(x_(2)-x_(1))^2+(y_(2)-y_(1))^2 }`

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

r =
`√((-1+5)^2+(5-4)^2)`

=
`√(4^2+1^2)`

=
`√(16+1)`

=
`√(17)` ⇒ r² = (
`√(17)` )² = 17

Then

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

Answer 2

x² + y² + 10x - 8y + 24 = 0

Explanation:

equation of a circle :-

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

• (h, k) is the center of the circle
• R is the radius of the circle

for this circle :-

• h = -5
• k = 4
• R = ?

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

`= > r = √(17)`

the radius if the circle is √17 units

__________________________________

finding the equation of the circle

=> (x + 5)² + (y - 4)² = (√17)²

=> x² + 25 + 10x + y² + 16 - 8y = 17

• as for any two variables a and b, (a + b)² = a² + b² + 2ab

=> x² + y² + 10x - 8y + 24 = 0