+2498 If a circle is centered at the origin and tangent to the line y=-x+4 , what is the area of the circle devided by pi ?
A) 2
B) 4
C) 8
D) 12
E) 16
+128407 Notice that a tangent line drawn from the center of the circle to the point of tangency will meet the given line at right angles..........thus, these two lines will have reciprocal slopes....
And the equation of the line through the origin will be
y = x
So......setting these two equations equal, we have
x = -x + 4 add x to both sides
2x = 4
x = 2
So, at the point of tangency, y = x, so y = 2
And the radius of such a circle centered at the origin will be
sqrt (2^2 + 2^2) = sqrt (8)
So....the area of this circle will be
A = pi [sqrt(8)]^2 = 8pi and dividing this by pi will produce a result of 8
Here's a graph of the situation ......https://www.desmos.com/calculator/g8eurlqsyo
+128407 Notice that a tangent line drawn from the center of the circle to the point of tangency will meet the given line at right angles..........thus, these two lines will have reciprocal slopes....
And the equation of the line through the origin will be
y = x
So......setting these two equations equal, we have
x = -x + 4 add x to both sides
2x = 4
x = 2
So, at the point of tangency, y = x, so y = 2
And the radius of such a circle centered at the origin will be
sqrt (2^2 + 2^2) = sqrt (8)
So....the area of this circle will be
A = pi [sqrt(8)]^2 = 8pi and dividing this by pi will produce a result of 8
Here's a graph of the situation ......https://www.desmos.com/calculator/g8eurlqsyo
