The line 4x-3y=-12 is tangent at the point (-3,0) and the line 3x+4y=16 is tangent at the point (4,1). find the equation of the circle.
solution) well you could first find the radius at (3,-4) since it is perpendicular to the tangent
that''s using two-points form: (for (0,0) and (3,-4) )
y + 4 = [(-4 -0) / (3 - 0 )]* (x - 3)
or, 3y + 12 = -4x + 12
or, y = (-4/3) x
let m be the slope of the tangent, then:
m * -4/3 = -1
or, m = 3/4
thus you use the equation:
y - y, = m (x - x,)
or, y + 4 = 3/4 (x - 3)
or, 4y + 16 = 3x - 9
or, 3x - 4y - 25 = 0
that''s the slope you need
of course, if you wanted to do it the shorter way, you could always use the formula:
ax + by = r^2
where (a,b) is the point (3,-4) and r is the radius squared, as in the circle''s equation x^2+y^2 = r^2, which is 25