Philip P. answered • 05/31/16

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Put the equation of the given line into slope-intercept form, y = mx+b, to find the slope m:

-7x + 24y = 625

24y = 7x + 625

y = (7/24)x + (625/24)

The slope is 7/24. A theorem of geometry states that

*if a radius meets a tangent line at the point of tangency, it will form a right angle (90*. Hence we want a line that is perpendicular to the given tangent line and which passes through the circle's center, (0,0). Two lines are perpendicular if their slopes are negative reciprocals of one another. Hence our radial line must have a slope of -24/7 and its equation is y = (-24/7) + b. Since the line passes through the origin (the circle's center), its y-intercept (b) is zero. So the full equation of the radial line is:^{o}) with the tangent line y = (-24/7)x

The point of tangency exists where the two lines intersect:

-(24/7)x = (7/24)x + (625/24)

Multiply both sides by 24*7 to get rid of the denominators:

(24)(-24x) = 7

^{2}x + 625*7 -576x = 49x + 4375

-625x = 4375

x = -7

To find y, plug x = -7 into the equation for either line:

y = (-24/7)x

y = (-24/7)*(-7)

y = 24

The point of tangency is located at

**(-7,24)**