Tangents are drawn from the point P(3, 4) to the ellipse `x^2/9+y^2/4=1` touching the ellipse at points A and B.

Options

A. (3, 0) and (0, 2)
B. `((-8)/(5),(2sqrt161)/(15))and (-(9)/(5),(8)/(5))`
C. `(-(8)/(5),(2sqrt161)/(15))and (0,2)`
D. `(3, 0) and (-(9)/(5),(8)/(5))`

Answer

D

Explanation

The equation of the chord of contact of tangents drawn from P (3, 4) to the given ellipse is
`(x)/(3)+y=1" "...(i)`
The ordinates of the point of intersection of this line and the ellipse are roots of the equation
`(y-1)^(2)+(y^(2))/(4)=1rArr5y^(2)-8y=0rArry=0,(8)/(5)`
Putting y = 0 in (i), we get `x = 3`
Putting ` y=(8)/(5)` in (i), we get `x=(9)/(5)`
Hence, the coordinates of A and B are (3, 0) and `(-(9)/(5),(8)/(5))` respectively.