A circle centered at the origin meets the line -7x+24y=625 tangentially. Find coordinates for the point of tangency.

Question

Please explain how to get the answer step by step.

Answer 1

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^o) with the tangent line.  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:

 

     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) = 7²x + 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)

 

 

Answer 2

Put the line in point-slope form:

    y = (7/24)x +(625/24)

The segment that is the radius from the center of the circle to the point of tangency will intersect this line at right angles at the point of tangency.

So this radial segment is along the line: (the slope is the negative reciprocal so that it is perpendicular, at right angles)

   y = (-24/7)x  which also goes through the circle center at (0,0) as desired.

To find the point of tangency we find where these two lines intersect, setting the y's equal:

   (7/24)x + (625/24) = (-24/7)x

Multiply all terms by 7*24 to get rid of the fractions:

    7*7*x + 625*7 = -24*24*x

    49x + 4375 = -576x

     4375 = -625x

      x = -7        (wow, it's a nice number!)

and then the y of tangency is:

     y = (-24/7)*(-7) = 24

So the point of tangency is  (-7,24)

 

(Sorry that I am not able to include a diagram...)

 

Answer 3

At the point of tangency the line is perpendicular to the radius/diameter.

The equation of the radius/diameter:

 

y = -24x/7 + b

0 = 0 + b

y = -24x/7

 

Determine the common solution of these two equations and you have the coordinates of the point of tangency.