The combined equation of the images of pair of lines given by `ax^2+2hxy+by^2=0` in the line mirror `y=0`, is

Options

A. `ax^2-2hxy+by^2=0`
B. `bx^2-2hxy+ay^2=0`
C. `bx^2+2hxy+ay^2=0`
D. None of these

Answer

A

Explanation

Let `y=m_1x_1y=m_2x` be the lines represented by `ax^2+2hxy+by^2=0`. Then, `m_1+m_2=-(2h)/(b)`and `m_1m_2=(a)/(b)`......(i)
Clearly , if `y=m_1x` makes an angle `theta_1` with `y=0`(X-axis), then its image in line mirror `y=0` makes an angle `-theta`, with X-axis . So, its equation is
`y-tan(-theta_1) x or y =-(tantheta_1)x or y=-m_1x`
Similarly , equation of the image of `y=m_2x in y=0` is `y=-m_2x`
Therefore , the contined equation of the images is `(y+m_1x)(y+m_2x)=0`
`rArr y^2+xy(m_1+m_2)+m_1m_2x^2=0` ltbr. `rArr y^2-(2h)/(b)xy+(a)/(b)x^2=0`
`therefore by^2-2hxy+ax^2=0`