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Question
Find the joint equation of pair of lines passing through the origin and perpendicular to the lines represented by ax2+ 2hxy + by2= 0
Solution
Given equation is ax2 + 2hxy + by2 = 0,
Let m1 and m2 be the slopes of the given lines
`∴ m_1+m_2=(-2h)/b and m_1m_2=a/b`
Since, the required lines are perpendicular to these lines
∴ slopes of the required lines are `-1/m_1 and -1/m_2`
Required lines also pass through the origin, therefore their equations are
`y=-1/m_1x and y=-1/m_2x`
∴ the joint equation of the lines is `(x+m_1y)(x+m_2y)=0`
`∴ x^2+(m_1+m_2)xy+m_1m_2y^2=0`
`∴ x^2+((-2h)/b)xy+(a/b)y^2=0`
`∴ bx^2-2hxy+ay^2=0`
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