109k views
0 votes
Question 19 A circle is described by the equation x^(2)-10x+y^(2)+8y=6. Identify the center and radius of the circle.

User Afeez Aziz
by
8.1k points

2 Answers

1 vote

Answer:

Explanation:

Sure. The equation of a circle with center (a, b) and radius r is given by

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

We can rewrite the given equation as

(x^2 - 10x + 25) + (y^2 + 8y + 16) = 49

Completing the square on both sides, we get

(x - 5)^2 + (y + 4)^2 = 7^2

Therefore, the center of the circle is (5, -4) and the radius is 7.

User Rohit Lal
by
7.7k points
1 vote

First, we have to rearrange this standard form of a circle's equation, which is (x-h)² + (y-k)² = r² where (h, k) are the center points of the circle and r is the radius of the circle.

To do this, we first group and complete the square for the x-terms and the y-terms. These terms are x^2 - 10x and y^2 + 8y in the equation.

Complete the square for x^2 - 10x:

In a quadratic expression ax^2+bx+c, the completion of the square requires adding and subtracting (b/2a)^2. Here, a = 1 and b = -10. Therefore, we get (-10/2*1)^2 = 25.
So, the completion for x^2 - 10x is (x^2 - 10x + 25 - 25), which is (x - 5)^2 - 25.

Complete the square for y^2 + 8y:

The completion requires adding and subtracting (b/2a)^2. Here, a = 1 and b = 8. Therefore, we get (8/2*1)^2 = 16.
So, completion for y^2 + 8y is (y^2 + 8y + 16 - 16), which is (y + 4)^2 - 16.

Substitute these terms back into the given equation:
x^2 - 10x+y^2+8y=6 becomes (x-5)^2 -25 + (y+4)^2 - 16 = 6, which simplifies to (x-5)^2 + (y+4)^2 = 9+25+16 =47

Comparing this with the standard form (x-h)² + (y-k)² = r², the center of the circle (h, k) is (5, -4) and the radius of the circle r is the square root of 47, which is approximately 6.85.

But in the given solution, radius of the circle is given as 3 which is bit confusing. Please verify the provided solution once. But the above explanation is how we transform a circle equation to its standard form and find out the circle's radius and center.

User Roberto Ferraris
by
7.8k points

No related questions found

Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.

9.4m questions

12.2m answers

Categories