Determine the center and the radius of the following circle:
x^2+y^2+6x-4y-5=0
I got for the radius 3(sqrt2)
I got for the center (-3,2)
Can you please tell me if I am correct?
The radius seems wierd.
The answers are correct.
The radius is sqrt(18) which you have correctly reduced to 3sqrt(2), well done!
To determine the center and radius of a circle given its equation, you can rewrite the equation in the standard form of a circle equation, which is:
(x - h)^2 + (y - k)^2 = r^2
where (h, k) represents the center of the circle, and r represents the radius.
Let's rewrite the given equation of the circle:
x^2 + y^2 + 6x - 4y - 5 = 0
To complete the square and convert it into the standard form, we can group the x-terms and the y-terms separately:
(x^2 + 6x) + (y^2 - 4y) = 5
Next, you need to add and subtract half of the coefficient of x (which is 6/2 = 3) squared inside the parentheses containing x, and also do the same for y. Similarly, we'll add and subtract half of the coefficient of y (which is -4/2 = -2) squared inside the parentheses containing y. So, the equation becomes:
(x^2 + 6x + 3^2) + (y^2 - 4y + (-2)^2) = 5 + 3^2 + (-2)^2
(x + 3)^2 + (y - 2)^2 = 14
Now, comparing the above equation with the standard form, we can conclude that the center of the circle is (-3, 2) (opposite signs of h and k) and the radius is the square root of 14. Therefore, your answer for the center is correct: (-3, 2).
However, the radius should be the square root of 14, not 3√2. So, your radius calculation is incorrect. The correct radius is √14, which is approximately 3.74.
So, to summarize:
Center of the circle: (-3, 2)
Radius of the circle: √14 (approximately 3.74)
Hope this helps!