Tangent to line x+2y=4 whose center (3,3) find the equation?

Question

analytic geometry

"CIRCLES"

 

Tangent to line x+2y+4 whose center (3,3) find the equation?

Answer 1

If x+2y = 4, then y = (4-x)/2 = (-1/2)x + 2 

 

If the line is tangent to the circle, the radius of the circle that contains the point of tangency must be perpendicular to the tangent line.  The slope of the tangent line is -0.5, so the radius containing the point of tangency has slope 2.    

 

The point of tangency can be expressed as T = (x, -0.5x+2).   Let

P = (3,3) be the center of the circle.

 

Slope PT = 2.  So, (-0.5x+2-3)/(x-3) = 2

 

                          (-0.5x-1)/(x-3) = 2

 

                           -0.5x - 1 = 2x - 6

 

                                  2.5x = 5

 

                                      x = 2

 

Thus, T = (2, 1) 

 

The length of the radius, r, of the circle is then the distance from (2,1) to (3,3).

 

    r = √[(3-(2))²+(3-1)²] = √5

 

Equation of circle:  (x-h)²+(y-k)² = r²

 

                           (x-3)²+(y-3)² = 5