If `4ab=3h^2`, then the ratio of the slopes of the lines represented by the equation `ax^2+2hxy+by^2=0` will be
(A) `sqrt(2):1`
(B) `sqrt(3):1`
(C) `2:1`
(D) `1:3`

Options

A. `sqrt(2):1`
B. `sqrt(3):1`
C. `2:1`
D. `1:3`

Answer

D

Explanation

Given `ax^2+2hxy+by^2=0` ...(i)
and `4ab=3h^2` ...(ii)
Let `m_1 and m_2` be the slopes of the line represented by Eq.(i) , we get
Then , `m_1+m_2=(-2h)/(b)` ....(iii)
and `m_1.m_2=(a)/(b)` .....(iv)
Now, `m_1-m_2=sqrt((m_1+m_2)^2-4m_1m_2)`
`=sqrt((4h^2)/(b^2)-(4a)/(b))=sqrt((4(h^2-ab))/b^2)`
`=sqrt((4(h^2-(3)/(4)h^2))/(b^2))`
`=sqrt((h^2)/(b^2))`
`rArr m_1-m_2=(h)/(b)` ...(iv)
Now ,`(m_1+m_2)/(m_1-m_2)=(b)/((h)/(b))=(-2)/(1)`
Applying componendo and dividendo, we get
`(2m_1)/(1m_2)=(-1)/(-3)rArr (m_1)/(m_2)=(1)/(3)`
`therefore m_1:m_3=1:3`