Answer:
(10, 0)
Step-by-step explanation:
A tangent line to a circle is a straight line that intersects the circle at exactly one point, known as the point of tangency. At the point of tangency, the tangent line is perpendicular to the radius of the circle drawn to that point.
Therefore, to find the point of tangency, first find the equation of the line that is perpendicular to y = (2/5)x - 4 and passes through the center of the circle (8, 5), and then find the point of intersection of the two lines.
When two lines are perpendicular, their slopes are negative reciprocals of each other. Therefore, since the slope of the tangent line is m = 2/5, the slope of the line perpendicular to the tangent line is m = -5/2. To find the equation of this perpendicular line, we can substitute this slope (m = -5/2) and the point (8, 5 into the point-slope form of a linear equation:
[tex]y-y_1=m(x-x_1)\\\\\\y-5=-\dfrac{5}{2}(x-8)\\\\\\y-5=-\dfrac{5}{2}x+20\\\\\\y=-\dfrac{5}{2}x+25[/tex]
To find the x-coordinate of the point of tangency, set the equations of the tangent line and perpendicular line equal to each other and solve for x:
[tex]\dfrac{2}{5}x-4=-\dfrac{5}{2}x+25\\\\\\\dfrac{2}{5}x+\dfrac{5}{2}x=25+4\\\\\\\dfrac{29}{10}x=29\\\\\\29x=290\\\\\\x=10[/tex]
To find the y-coordinate of the point of tangency, substitute x = 10 into y = (2/5)x - 4:
[tex]y=\dfrac{2}{5}(10)-4\\\\\\y=\dfrac{20}5{}-4\\\\\\y=4-4\\\\\\y=0[/tex]
Therefore, the point of tangency is:
[tex]\LARGE\boxed{\boxed{(10,0)}}[/tex]
