Write the equation of the circle satisfying the center (3,4) and tangent to 2x - y + 5 = 0
+23246 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
+23246 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
