Question

Find equations of both the tangent lines to the ellipse

Answer 1

X²+4y²=36 / :36
x²/36 + y² / 4=1
a²=36. b²=9, where:a- semi-major axis, b - semi-minor axis
If the tangent line passes through point (12, 3 ) then the line equation is:
3 = 12 m + n  (m=?, n=?),      n = 3 - 2 m ( for the substitution )
Formula for tangent lines to the ellipse:
a²m² + b² = n²
36 m² + 9 = ( 3 - 2 m )²
36 m² + 9 = 9 - 72 m + 144 m²
-108 m² + 72 m = 0
m ( -180 m + 72 ) = 0,            m 1= 0,  m 2 =72/108 = 2/3
n 1=3 - 0 = 3,   n 2= 3 - 12 * 2/3 = 3 - 8 =-5
Equations of the tangent lines are:
y = 3 x       and     y= 2/3 x - 5