Final answer:
The question involves finding the x-intercepts of a circle with a known center and a point it passes through. By calculating the radius with the distance formula and then setting up the circle equation, the student can solve for x when y equals zero to find the x-intercepts.
Step-by-step explanation:
The student is asking about finding the x-intercepts of a circle with given center and a point it passes through. To find the x-intercepts of the circle, we need to use the distance formula to calculate the radius of the circle from the center -5,7 to the point 5,-1. Once the radius is known, we can set up the circle equation and solve for points where y equals zero, as this condition is true when the circle intersects the x-axis.
The distance formula is d = \sqrtx_2 - x_1^2 + y_2 - y_1^2 \. Using the points -5,7 and 5,-1, we get \ d = \sqrt5 - -5^2 + -1 - 7)^2 = \sqrt1100+ 64 = \sqrt164 \. Hence, the radius of the circle is the square root of 164.
Next, we set up the circle's equation using the radius and the center's coordinates: \x - -5^2 + y - 7^2 = \sqrt164^2 \, which simplifies to \ x + 5^2 + y - 7^2 = 164 \. To find the x-intercepts, substitute y = 0 into the equation and solve for x.
After simplifying, we will get the x-coordinates of the x-intercepts of the circle.