Select the correct answer.

Segment [tex]$KL$[/tex] is a diameter of circle [tex]$Q$[/tex]. Circle [tex]$Q$[/tex] is represented by the equation [tex]$(x - 11)^2 + (y + 15)^2 = 7$[/tex]. What is [tex]$KL$[/tex]?

A. 98
B. 14
C. [tex]$2\sqrt{7}$[/tex]
D. [tex]$\sqrt{14}$[/tex]



Answer :

To determine the length of segment [tex]\( KL \)[/tex], which is the diameter of the circle represented by the equation [tex]\((x - 11)^2 + (y + 15)^2 = 7\)[/tex], we need to interpret this equation in terms of its standard form.

The standard form of a circle's equation is:
[tex]\[ (x - h)^2 + (y - k)^2 = r^2 \][/tex]
where [tex]\((h, k)\)[/tex] is the center of the circle and [tex]\(r\)[/tex] is the radius.

From the equation [tex]\((x - 11)^2 + (y + 15)^2 = 7\)[/tex], we can observe the following:
- The center of the circle [tex]\((h, k)\)[/tex] is [tex]\((11, -15)\)[/tex].
- The radius squared [tex]\(r^2\)[/tex] is [tex]\(7\)[/tex].

To find the radius [tex]\(r\)[/tex] of the circle, we take the square root of [tex]\(7\)[/tex]:
[tex]\[ r = \sqrt{7} \][/tex]

The diameter [tex]\(KL\)[/tex] of the circle is twice the radius:
[tex]\[ KL = 2 \times r = 2 \times \sqrt{7} \][/tex]

Thus, [tex]\(KL = 2 \sqrt{7}\)[/tex].

So, the correct answer is:
C. [tex]\(2 \sqrt{7}\)[/tex]