Question

how do you rewrite the equation x² + y² + 6x - 8y - 11 = 0 in standard form? a. (x - 3)² + (y - 4)² = 36 b. (x + 3)² + (y - 4)² = 36 c. (x + 3)² + (y + 4)² = 36 d. (x - 3)² + (y + 4)² = 36 which of the following represents a circle with its center at the origin and a radius of 10? a. (x + 10)² + (y + 10)² = 100 b. x² + y² = 10 c. x² - y² = 100 d. x² + y² = 100 the equation of a circle is (x + 3)² + (y - 5)² = 16. what is the distance from the center to the point (1, 5)? a. 6 b. 8 c. 2 d. 4

Explanation:

Step1: Rewrite the first - given equation to standard form

The general equation of a circle is \((x - a)^2+(y - b)^2=r^2\), and we complete the square for \(x^{2}+y^{2}+6x - 8y-11 = 0\).
For the \(x\) - terms: \(x^{2}+6x=(x + 3)^{2}-9\).
For the \(y\) - terms: \(y^{2}-8y=(y - 4)^{2}-16\).
So, \(x^{2}+y^{2}+6x - 8y-11=(x + 3)^{2}-9+(y - 4)^{2}-16-11=0\).
Simplify to get \((x + 3)^{2}+(y - 4)^{2}=36\).

Step2: Determine the equation of a circle centered at the origin

The equation of a circle with center \((0,0)\) and radius \(r\) is \(x^{2}+y^{2}=r^{2}\). Given \(r = 10\), the equation is \(x^{2}+y^{2}=100\).

Step3: Find the center of the given circle and calculate the distance

The equation of the circle is \((x + 3)^{2}+(y - 5)^{2}=16\), so the center is \((-3,5)\).
The distance \(d\) between two points \((x_1,y_1)\) and \((x_2,y_2)\) is given by the distance formula \(d=\sqrt{(x_2 - x_1)^{2}+(y_2 - y_1)^{2}}\). Here, \((x_1,y_1)=(-3,5)\) and \((x_2,y_2)=(1,5)\).
\(d=\sqrt{(1-(-3))^{2}+(5 - 5)^{2}}=\sqrt{(4)^{2}+0^{2}}=4\).

Answer:

1. B. \((x + 3)^{2}+(y - 4)^{2}=36\)
2. D. \(x^{2}+y^{2}=100\)
3. D. 4