Given the circle (x – 6)2 + (y + 2)2 = 9, determine if the line with an equation of 3x – 2y = 12 is a tangent

Question

Given the circle (x – 6)2 + (y + 2)2 = 9, determine if the line with an equation of 3x – 2y = 12 is a tangent to the circle, a secant to the circle, or neither.

tangent
secant
neither

Answer 1

2y = 3x - 12
y = (3/2) x - 6
(x - 6) ² + [(3/2)x - 4) ] ² = 9
(x - 6)² + (9/4)x² - 12x + 16 = 9
x² - 12x + 36 + (9/4)x² - 12x + 7 = 0
(13/4)x² - 24x + 43 = 0
13x² - 96x + 172 = 0
Roots are not equal so line is NOT a tangent.
x = [96 ±√(9216- 8944] / 26
x = [96 ±√272 ] / 26
x = [96 ± 16.5 ] / 26
Real roots so line cuts circle in two places
ie secant