Given x2 + y2 -3mx - 8y + 16=0 as an equation of a circle centered at ( 3, n ). Find the value of m and n. Hence the length of its radius

Question

Given x² + y² -3mx - 8y + 16=0 as an equation of a circle centered at ( 3, n ). Find the value of m and n. Hence the length of its radius

Answer 1

The generic equation for a circle at (h, k) is (x - h)² + y - k)² = r²

For x² + y² - 3mx - 8y + 16 = 0 we can separate the variables:

(x² - 3mx + ) + (y² - 8y + 16) = 0

(x² - 3mx + ) + (y - 4)² = 0

So, we know that n = 4.

Since the x coordinate of the center is 3, we know that the square associated with the "x" is (x - 3)² meaning it equates to x² - 6x + 9 and since we have "(x² - 3mx + )" in our equation, then m must equal 2 ("-3m = -6").

Replacing m with 2, we get:

(x² - 6x + ) + (y - 4)² = 0

Adding 9 to both sides we get:

(x² - 6x + 9) + (y - 4)² = 9

(x - 3)² + (y - 4)² = 3² meaning the radius is 3

So m = 2, n = 4 and r = 3

Answer 2

We are given an equation of a circle x² + y² - 3mx - 8y + 16 = 0 with center (3, n)

The center-radius form of the equation of a circle is (x – h)² + (y – k)² = r² with center (h,k) and radius r

In this case, we know that (x - 3)² + (y - n)² = r²

which expands to x² - 6x + 9 + y² - 2yn + n² = r²

and can be rearranged to look like our original equation as

x² + y² - 6x - 2yn + n² - r² + 9 = 0

x² + y² - 3mx - 8y + 16 = 0

comparing terms in the two equations we can say:

(1) -3mx = -6x => m = 2

(2) -2yn = -8y => n = 4

(3) n² - r² + 9 = 16

=> r² = n² + 9 - 16 => r² = (4)² + 9 - 16 = 9 => r = √9 = +3 or - 3

hence, our circle has the equation (x - 3)² + (y - 4)² = 9 with center at (3, 4) and radius of 3

to double-check substitute these values in the circle equation (x – h)² + (y – k)² = r²

giving us (x - 3)² + (y - 4)² = (3)²

=> x² - 6x + 9 + y² - 8y + 16 = 9

=> x² + y² - 6x - 8y + 9 + 16 = 9

=> x² + y² - 6x - 8y + 16 = 0

and x² + y² - 3mx - 8y + 16 = 0 given equation

Answer 3

The standard equation of a circle is (x - h)² + (y - k)² = r² where the center is (h, k) and r is the radius. Plugging in (3, n) for the center gives us (x - 3)² + (y - n)² = r². Expand the (x - 3)² and (y - n)² terms and then compare what you get to the original equation. That should help you figure out m, n, and r.