Question

The equation of a circle is shown. What is the radius? 22 + y2 - 10x + 8y + 16 = 0​

Answer 1

radius = 5

Explanation:

Given

• x² + y² - 10x + 8y + 16 = 0

Using completing the square method

• x² - 10x + 25 - 25 + y² + 8y + 16
• (x + 5)² + (y + 4)² - 25 = 0
• (x + 5)² + (y + 4)² = 25

Standard form for equation of a circle

• (x - h)² + (y - k)² = r², where r is the radius

On comparison to our equation in the question :

• r² = 25
• radius = 5 [as distances cannot be -ve]

Answer 2

radius = 5

Explanation:

Question

`x^2+y^2-10x+8y+16=0`

The equation of a circle is shown. What is the radius?

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Equation of a circle

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

(where (a, b) is the center and r is the radius)

Given equation:

`x^2+y^2-10x+8y+16=0`

Collect like terms:

`\implies x^2-10x+y^2+8y+16=0`

Subtract 16 from both sides:

`\implies x^2-10x+y^2+8y=-16`

Complete the square for both variables.

Add 25 to both sides for x. Add 16 to both sides for y.

`\implies x^2-10x+25+y^2+8y+16=-16+25+16`

`\implies (x^2-10x+25)+(y^2+8y+16)=25`

Factor the two variables:

`\implies (x-5)^2+(y+4)^2=25`

Therefore:

• center of the circle = (5, -4)
• radius of the circle = √25 = 5