Find the unique point that is the center of a circle passing through (2,5) and touching the line y = -7
The locus of the center of a circle has to be found that passes through (2, 5) and touches the line y = -7. The line y = -7 is a tangent of the circle.
The general equation of a circle is x^2 + y^2 + ax + by + c = 0
As the circle passes through (2, 5)
=> 4 + 25 + 2a + 5b + c = 0
=> 2a + 5b + c + 29 = 0 ...(1)
At y = -7, the equation has only one root
x^2 + 49 + ax - 7b + c = 0 has only one root, this gives:
a^2 = 4*1*(c + 49 - 7b) ...(2)
From (1) and (2) it can be seen that there are 3 variables but only 2 equations.
This allows for an infinite number of solutions for a, b and c.
The center of the circle satisfying the given conditions is not a unique point.