Question

X^2+y^2-16x+8y=11 R-?

Answer 1

Сложим в квадраты разности и суммы

$x^2+y^2-16x+8y-11=0\\\\ x^2-16x=(x-8)^2-64\\ y^2+8y=(y+4)^2-16\\ -11=-11$

Теперь сложим все уравнения

$x^2-16x+y^2+8y-11=(x-8)^2-64+(y+4)^2-16-11=0\\ (x-8)^2+(y+4)^2=91\\ R= \sqrt{91}$

Answer 2

The given equation is in the form of a standard circle equation. By completing the square, we can determine the radius and center of the circle.

Let's rewrite the equation: x^2 + y^2 - 16x + 8y = 11

To complete the square for the x-terms, we need to add and subtract the square of half the x-coefficient (which is -16/2 = -8): (x^2 - 16x) + (y^2 + 8y) = 11

Now, for the y-terms: (x^2 - 16x + 64) + (y^2 + 8y + 16) = 11 + 64 + 16 (x^2 - 16x + 64) + (y^2 + 8y + 16) = 91

Simplifying further: (x - 8)^2 + (y + 4)^2 = 91

Comparing this equation with the standard form of a circle: (x - h)^2 + (y - k)^2 = r^2

We can see that the center of the circle is at the point (h, k) = (8, -4), and the radius squared is r^2 = 91. Taking the square root of 91 will give us the radius of the circle.

Therefore, the radius of the given circle is √91 (approximately 9.54).