Question

find the distance between the pair of points.
(5 - √7, - 3) and (- 4 - √7, 2)
16
3
8
4
question 10
write the standard form of the equation of the circle with the given center and radius.
(7, - 9); √11
(x - 9)²+(y + 7)² = 121
(x - 7)²+(y + 9)² = 11
(x + 7)²+(y - 9)² = 11
(x + 9)²+(y - 7)² = 121

Explanation:

Step1: Recall distance formula

The distance formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$. Let $(x_1,y_1)=(5\sqrt{2}, - 3)$ and $(x_2,y_2)=(4\sqrt{2},2)$.

Step2: Calculate differences

$x_2 - x_1=4\sqrt{2}-5\sqrt{2}=-\sqrt{2}$ and $y_2 - y_1=2-( - 3)=5$.

Step3: Apply formula

$d=\sqrt{(-\sqrt{2})^2 + 5^2}=\sqrt{2 + 25}=\sqrt{27}=3\sqrt{3}$ (There seems to be an error in the problem - setup as the provided answer - choices don't match this result. Let's assume the points are $(5, - 3)$ and $(4,2)$). Then $x_2 - x_1=4 - 5=-1$ and $y_2 - y_1=2-( - 3)=5$. So $d=\sqrt{(-1)^2+5^2}=\sqrt{1 + 25}=\sqrt{26}$ (still not in the choices). If we assume the points are $(5,-3)$ and $(4, - 1)$: $x_2 - x_1=4 - 5=-1$ and $y_2 - y_1=-1-( - 3)=2$. Then $d=\sqrt{(-1)^2+2^2}=\sqrt{1 + 4}=\sqrt{5}$ (not in choices). Let's use the correct - way for the circle problem.
The standard form of the equation of a circle is $(x - h)^2+(y - k)^2=r^2$, where $(h,k)$ is the center of the circle and $r$ is the radius.

Step1: Identify center and radius from the problem

Assume the center of the circle is $(h,k)=(7, - 9)$ and radius $r=\sqrt{11}$.

Step2: Write the equation

The equation of the circle is $(x - 7)^2+(y+9)^2 = 11$.

Answer:

For the first problem (distance between points), the problem may have incorrect data as the calculated results don't match the given choices.
For the circle - equation problem: $(x - 7)^2+(y + 9)^2=11$ corresponds to the second - underlined option in the circle - equation part of the problem.