Answer:
Option C.
The distance from the point to the circle is 6 units
Step-by-step explanation:
we know that
The distance between the circle and the point will be the difference of the distance of the point from the center of circle and the radius of the circle
step 1
Find the center and radius of the circle
we have
[tex]x^{2} +y^{2}-8y=0[/tex]
Convert to radius center form
Complete the square
[tex]x^{2} +(y^{2}-8y+16)=0+16[/tex]
Rewrite as perfect squares
[tex]x^{2} +(y-4)^{2}=16[/tex]
so
The center is the point (0,4)
The radius is
[tex]r^{2}=16\\r=4\ units[/tex]
step 2
Find the distance of the point from the center of circle
the formula to calculate the distance between two points is equal to
[tex]d=\sqrt{(y2-y1)^{2}+(x2-x1)^{2}}[/tex]
we have
(6,-4) and (0,4)
substitute the values
[tex]d=\sqrt{(4+4)^{2}+(0-6)^{2}}[/tex]
[tex]d=\sqrt{(8)^{2}+(-6)^{2}}[/tex]
[tex]d=\sqrt{100}\ units[/tex]
[tex]d=10\ units[/tex]
step 3
Find the difference of the distance of the point from the center of circle and the radius of the circle
[tex]10\ units-4\ units=6\ units[/tex]
therefore
The distance from the point to the circle is 6 units
see the attached figure to better understand the problem
