Find the equation of pair of tangents drawn from point (4, 3) to the hyperbola `(x^(2))/(16)-(y^(2))/(9)=1`. Also, find the angle between the tangents.

Explanation

Equation of pair of tengents is `T^(2)=SS_(1)`.
`therefore" "((4x)/(16)-(3y)/(9)-1)^(2)=((x^(2))/(16)-(y^(2))/(9)-1)(-1)`
`rArr" "(x^(2))/(16)+(y^(2))/(9)+1-(xy)/(6)-(x)/(2)+(2y)/(3)=-(x^(2))/(16)+(y^(2))/(9)+1`
`rArr" "(x^(2))/(8)-(xy)/(6)-(x)/(2)+(2y)/(3)=0`
`rArr 3x^(2)-4xy-12x+16y=0,` which is requird equation of pair of tangents.
Comparing it with standard second-degree equation, we have
`a=3,b=0 and h=-2`
`therefore` Angle between tangents,
`theta=tan^(-1).(2sqrt(h^(2)-ab))/(|a+b|)`
`=tan^(-1).(2sqrt((-2)^(2)-(3)(0)))/(|3+0|)`
`=tan^(-1).(4)/(3)`