Question

Find the equations of the tangents to the ellipse: 5x² + 9y² = 45 which are ⊥ to the line 3x +
2y + 1 = 0.



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Answer 1

Answer:

Step-by-step explanation:

The equations of tangent to the ellipse x²/a² + y²/b² = 1 in terms of slope m are y = mx±√(a²m² + b²)  

The equation of ellipse is 5x² + 9y² = 45

=> 5x²/45 + 9y²/45 = 1

=> x²/9 + y²/5 = 1

Comparing this with x²/a² + y²/b² = 1, we get

a² = 9, b² = 5

slope of line 3x + 2y + 1 = 0

=> 2y = - 3x - 1

=> y = (-3/2)x + (-1/2)

comparing this with y = mx + c, we get

m = -3/2.

Since the tangent is perpendicular to this line, it slope m = 2/3.

Substituting these values back in equation of tangents  y = mx±√(a²m² + b²)

y = (2/3)x±√[(9)(2/3)² + 5]

  = 2x/3 ±√[9*4/9 + 5]

  = 2x/3 ±√[4+5]

   =2x/3 ±√(9)

   = 2x/3 ±3

=> 3y = 2x ± 9.

=> 2x - 3y = ± 9