The circles $x^2+y^2=9$ and $x^2+y^2-12 y+27=0$ touch each other then the equation of the common tangent is:
a) $x-3 y+1=0$
b) $y=3$
c) $x+y=2$
d) $x=4$
Answer (1)
Determine the centers and radii of both circles from their equations.
Calculate the distance between the centers and verify that the circles touch externally.
Recognize that the common tangent is perpendicular to the line connecting the centers.
Find the midpoint between the centers, which lies on the common tangent, and deduce the equation of the tangent: y = 3 .
Explanation
Problem Analysis We are given two circles: x 2 + y 2 = 9 and x 2 + y 2 − 12 y + 27 = 0 . We need to find the equation of the common tangent.
Circle Properties The first circle has center ( 0 , 0 ) and radius r 1 = 9 = 3 . The second circle can be rewritten as x 2 + ( y 2 − 12 y + 36 ) = 36 − 27 , which simplifies to x 2 + ( y − 6 ) 2 = 9 . Thus, the second circle has center ( 0 , 6 ) and radius r 2 = 9 = 3 .
Distance Between Centers The distance between the centers of the two circles is d = ( 0 − 0 ) 2 + ( 6 − 0 ) 2 = 0 + 36 = 6 . Since r 1 + r 2 = 3 + 3 = 6 , the circles touch each other externally.
Tangent Direction The common tangent will be perpendicular to the line connecting the centers of the circles. The line connecting the centers is the y-axis (the line x = 0 ). Therefore, the common tangent must be a horizontal line, which has the form y = c for some constant c .
Tangent Point Since the circles touch each other externally, the common tangent passes through the point where the two circles touch. This point is the midpoint of the segment connecting the centers of the circles. The midpoint is ( 2 0 + 0 , 2 0 + 6 ) = ( 0 , 3 ) .
Tangent Equation Since the common tangent is a horizontal line and passes through the point ( 0 , 3 ) , its equation is y = 3 .
Verification Let's verify that y = 3 is tangent to both circles. For the first circle, x 2 + 3 2 = 9 , so x 2 = 0 , which means x = 0 . Thus, the line y = 3 is tangent to the first circle at ( 0 , 3 ) . For the second circle, x 2 + ( 3 − 6 ) 2 = 9 , so x 2 + ( − 3 ) 2 = 9 , which means x 2 = 0 , so x = 0 . Thus, the line y = 3 is tangent to the second circle at ( 0 , 3 ) .
Final Answer Therefore, the equation of the common tangent is y = 3 .
Examples
Understanding common tangents is crucial in engineering, especially when designing systems involving gears or pulleys. Ensuring proper contact and smooth operation requires precise calculations of tangent lines to circular components. For example, in a two-gear system, the common tangent at the point of contact determines the direction of force transmission, influencing the efficiency and stability of the mechanical system.
