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Write the equation of the circle satisfying the center (3,4) and tangent to 2x - y + 5 = 0

 Feb 28, 2020

Best Answer 

 #1
avatar+21951 
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Plan:  find the point where the circle is tangent to the line by finding the line that passes through the point (3,4) and is perpendicular to the line 2x - y + 5 = 0.

 

Find the slope of 2x - y + 5 = 0   --->   2x + 5 = y   --->   its slope is 2

The slope of the line perpendicular to this line is -1/2 = -0.5

 

Find the equation of the line through (3,4) with a slope of -0.5   --->   y - 4  =   \-0.5(x - 3)

                                                                                                    --->  2y - 8  =  -x + 3

                                                                                                    --->        2y  =  -x + 11

 

Find where the lines   y  =  2x + 5   and   2y  =  -x + 11   intersect.

                                  2y  =  4x + 10  and  2y  =  -x + 11   

Setting these equations equal to each other:

--->         4x + 10  =  -x + 11     --->     5x  =  1     --->     x  =  0.2

Finding y:  y  =  2x + 5   --->     y  =  2(0.2) + 5    --->     y  =  5.4

Therefore, the point of tangency of the circle and the line is  (0.2, 5.4)

 

Find the distance from the center of the circle to the point of tangency:

   Distance  =  sqrt[ (3 - 0.2)2  +  (4 - 5.4)2 ]   =  sqrt(9.8)     <--- this is the length of the radius

 

Equation of the circle:  (x - 3)2 + (y - 4)2  =  9.8

 Feb 28, 2020
 #1
avatar+21951 
+1
Best Answer

Plan:  find the point where the circle is tangent to the line by finding the line that passes through the point (3,4) and is perpendicular to the line 2x - y + 5 = 0.

 

Find the slope of 2x - y + 5 = 0   --->   2x + 5 = y   --->   its slope is 2

The slope of the line perpendicular to this line is -1/2 = -0.5

 

Find the equation of the line through (3,4) with a slope of -0.5   --->   y - 4  =   \-0.5(x - 3)

                                                                                                    --->  2y - 8  =  -x + 3

                                                                                                    --->        2y  =  -x + 11

 

Find where the lines   y  =  2x + 5   and   2y  =  -x + 11   intersect.

                                  2y  =  4x + 10  and  2y  =  -x + 11   

Setting these equations equal to each other:

--->         4x + 10  =  -x + 11     --->     5x  =  1     --->     x  =  0.2

Finding y:  y  =  2x + 5   --->     y  =  2(0.2) + 5    --->     y  =  5.4

Therefore, the point of tangency of the circle and the line is  (0.2, 5.4)

 

Find the distance from the center of the circle to the point of tangency:

   Distance  =  sqrt[ (3 - 0.2)2  +  (4 - 5.4)2 ]   =  sqrt(9.8)     <--- this is the length of the radius

 

Equation of the circle:  (x - 3)2 + (y - 4)2  =  9.8

geno3141 Feb 28, 2020

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