Answer :
To find out which equation represents a circle that contains the point [tex]$(-2, 8)$[/tex] and has a center at [tex]$(4, 0)$[/tex], we need to determine the radius of the circle first and then see which of the given equations matches this information.
### Step 1: Calculate the radius
The radius [tex]\( r \)[/tex] of the circle can be found using the distance formula between the point [tex]$(-2, 8)$[/tex] and the center of the circle [tex]$(4, 0)$[/tex].
The distance formula is:
[tex]\[ r = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
Substituting the given points [tex]$(x_1, y_1) = (-2, 8)$[/tex] and [tex]$(x_2, y_2) = (4, 0)$[/tex],
[tex]\[ r = \sqrt{(4 - (-2))^2 + (0 - 8)^2} = \sqrt{(4 + 2)^2 + (-8)^2} = \sqrt{6^2 + (-8)^2} = \sqrt{36 + 64} = \sqrt{100} = 10 \][/tex]
So, the radius [tex]\( r \)[/tex] is [tex]$10$[/tex].
### Step 2: Write the equation of the circle
The standard form of the equation of a circle with center [tex]$(h, k)$[/tex] and radius [tex]$r$[/tex] is:
[tex]\[ (x - h)^2 + (y - k)^2 = r^2 \][/tex]
Substituting [tex]\( h = 4 \)[/tex], [tex]\( k = 0 \)[/tex], and [tex]\( r = 10 \)[/tex],
[tex]\[ (x - 4)^2 + y^2 = 10^2 = 100 \][/tex]
### Step 3: Verify which given equation matches
From the options provided, we need to identify which equation matches the calculated circle's equation.
1. [tex]$(x - 4)^2 + y^2 = 100$[/tex]
2. [tex]$(x - 4)^2 + y^2 = 10$[/tex]
3. [tex]$x^2 + (y - 4)^2 = 10$[/tex]
4. [tex]$x^2 + (y - 4)^2 = 100$[/tex]
The equation of the circle derived above is:
[tex]\[ (x - 4)^2 + y^2 = 100 \][/tex]
Comparing this with the given options, the first option [tex]$(x - 4)^2 + y^2 = 100$[/tex] matches perfectly.
### Conclusion
The equation that represents a circle containing the point [tex]$(-2, 8)$[/tex] and having a center at [tex]$(4, 0)$[/tex] is:
[tex]\[ (x - 4)^2 + y^2 = 100 \][/tex]
### Step 1: Calculate the radius
The radius [tex]\( r \)[/tex] of the circle can be found using the distance formula between the point [tex]$(-2, 8)$[/tex] and the center of the circle [tex]$(4, 0)$[/tex].
The distance formula is:
[tex]\[ r = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
Substituting the given points [tex]$(x_1, y_1) = (-2, 8)$[/tex] and [tex]$(x_2, y_2) = (4, 0)$[/tex],
[tex]\[ r = \sqrt{(4 - (-2))^2 + (0 - 8)^2} = \sqrt{(4 + 2)^2 + (-8)^2} = \sqrt{6^2 + (-8)^2} = \sqrt{36 + 64} = \sqrt{100} = 10 \][/tex]
So, the radius [tex]\( r \)[/tex] is [tex]$10$[/tex].
### Step 2: Write the equation of the circle
The standard form of the equation of a circle with center [tex]$(h, k)$[/tex] and radius [tex]$r$[/tex] is:
[tex]\[ (x - h)^2 + (y - k)^2 = r^2 \][/tex]
Substituting [tex]\( h = 4 \)[/tex], [tex]\( k = 0 \)[/tex], and [tex]\( r = 10 \)[/tex],
[tex]\[ (x - 4)^2 + y^2 = 10^2 = 100 \][/tex]
### Step 3: Verify which given equation matches
From the options provided, we need to identify which equation matches the calculated circle's equation.
1. [tex]$(x - 4)^2 + y^2 = 100$[/tex]
2. [tex]$(x - 4)^2 + y^2 = 10$[/tex]
3. [tex]$x^2 + (y - 4)^2 = 10$[/tex]
4. [tex]$x^2 + (y - 4)^2 = 100$[/tex]
The equation of the circle derived above is:
[tex]\[ (x - 4)^2 + y^2 = 100 \][/tex]
Comparing this with the given options, the first option [tex]$(x - 4)^2 + y^2 = 100$[/tex] matches perfectly.
### Conclusion
The equation that represents a circle containing the point [tex]$(-2, 8)$[/tex] and having a center at [tex]$(4, 0)$[/tex] is:
[tex]\[ (x - 4)^2 + y^2 = 100 \][/tex]
