The equations of the tangents to the ellpise `4x^(2) +3y^(2)=5,` which are incrlined at `60^(@)` to the axis of x are

Options

A. `y=sqrt(3)x+-sqrt((65)/(12))`
B. `y=sqrt(3)x+-sqrt((12)/(65))`
C. `y=(x)/(sqrt(3))+-sqrt((12)/(65))`
D. none of these

Answer

A

Explanation

the equations of the tangents to `(x^(2))/(a^(2))+(y^(2))/(b^(2))=1` are given by `y=mx+- sqrt(a^(2)m^(2)+b^(2))`, where m is the slope of the tangent.
for the ellipse` 4x^(2)+3y^(2) =5,` we have
`a=(sqrt(5))/(2) ,b=(sqrt(5))/(sqrt(3)),m= tan 60^(@) =sqrt(3)`
so the equation of the tangent are
`y=sqrt(3) x+- sqrt((5)/(4)xx3+(5)/(3) )implies y=sqrt(3) x+- sqrt((65)/(12))`