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Question

(A) The equations of the required lines passing through the origin with slopes $m_1 = \sqrt{3}+1$ and $m_2 = \sqrt{3}-1$ are $y = m_1 x$ and $y = m_2

Vedclass · Accepted Answer

$m^{2}-2 \sqrt{3} m+2=0$

Vedclass · Answer

(A) The equations of the required lines passing through the origin with slopes $m_1 = \sqrt{3}+1$ and $m_2 = \sqrt{3}-1$ are $y = m_1 x$ and $y = m_2 x$.Since the auxiliary equation for a pair of lines $y = m_1 x$ and $y = m_2 x$ is $(m - m_1)(m - m_2) = 0$,we substitute the given slopes:$(m - (\sqrt{3}+1))(m - (\sqrt{3}-1)) = 0$$m^2 - m(\sqrt{3}-1) - m(\sqrt{3}+1) + (\sqrt{3}+1)(\sqrt{3}-1) = 0$$m^2 - m(\sqrt{3}-1 + \sqrt{3}+1) + ((\sqrt{3})^2 - 1^2) = 0$$m^2 - m(2\sqrt{3}) + (3 - 1) = 0$$m^2 - 2\sqrt{3}m + 2 = 0$

The auxiliary equation of the lines passing through the origin and having slopes $\sqrt{3}+1$ and $\sqrt{3}-1$ is

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