Tangents are drawn from the point P(3,4) to the ellipse x²/9 + y²/4 = 1 touching the ellipse at points A and B. The coordinates of A and B are?

Given equation of ellipse is x²/9 + y²/4 = 1 (i)
Equation of chord AB is xx₁/a² + yy₁/b² = 1
3x/9 + 4y/4 = 1
x + 3y = 3
x = 3(-y + 1)
Now, from (i)
9(-y+1)²/9 + y²/4 = 1
y² - 6y + 1 + y²/4 = 1
5y²/4 - 6y = 0
y(5y/4 - 6) = 0
y = 0 or 8/5
=> x = 3 when y = 0
x = 3 - 3(8/5) when y = 8/5
=> x = -9/5
(x,y) = (3,0) or (-9/5, 8/5)