Question

Please explain your work still don’t understand

Answer 1

Answer:

• The radius is 5√2.
• The center is (-3, 4).

Step-by-step explanation:

It can be helpful to understand what the square of a binomial looks like:

  (a +b)² = a² +2ab +b²

The middle term on the right is twice the product of the terms in the original binomial on the left.

Here, we want to use this relationship to find "b" when we're given "2ab". We recognize that "b" is half the coefficient of "a" in 2ab.

Choosing a value for b² to turn the sum (a² +2ab) into the trinomial (a² +2ab +b²) is called "completing the square" because that trinomial can now be written as the square (a+b)².

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The standard form equation of a circle with center (h, k) and radius r is ...

  (x -h)² +(y -k)² = r²

In order to find the center and radius of the circle from the given equation, you're expected to rewrite the equation in this form. You do that by "completing the square" for both x-terms and y-terms.

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Given

  x² +y² +6x -8y -25 = 0

Regrouping, we have ...

  (x² +6x) +(y² -8y) = 25

Adding the squares of half of 6 and half of -8, we can write this as ...

  (x² +6x +3²) +(y² -8y +(-4)²) = 25 +3² +(-4)²

And writing the trinomials as squares gives us ...

  (x +3)² +(y -4)² = 50 = (5√2)²

Comparing this to the standard form equation above, we see that ...

  (h, k) = (-3, 4)

  r = 5√2

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The radius is 5√2.

The center is (-3, 4).

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The attachment shows that the original equation draws a circle with center (-3, 4) and through points that are 5 units horizontally and vertically from the center, such as the point (2, -1). That is, the radius is 5√2.