+0  
 
-1
30
1
avatar

A circle centered at the origin is tangent to the line x-2y+25=0. What is the area of the circle?  Find the coordinates of the point of tangency.

 Apr 27, 2022
 #1
avatar+122385 
+1

Distance from origin  to  the line  =

 

 l 0 - 2(0) +  25  l   / sqrt  (1^2  + 2^2)   =

 

25 / sqrt  5  =

 

5sqrt (5)

 

This is the radius  of the  circle

 

Area =   pi ( 5sqrt (5) )^2    =    125 pi  units ^2

 

The  given line will have a slope of    1/2

 

The  equation  of   a  line  perpendicular to this  passing through the origin  has  the equation

 

y   =  -2x    (1)

 

The original line  can be written as

 

y  =  (1/2)x + 25/2     (2)

 

Set  (1)  =  (2)

 

-2x  = (1/2)x  + 25/2

 

-4x  = x  +  25

 

-5x  =  25

 

x = - 5    y =   -2(-5) =  10

 

Point of tangency =   (-5 , 10)

 

 

cool cool cool

 Apr 27, 2022

28 Online Users