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Question

Find the angle between the line given by the
equation $$\lambda y^{2}+(1-\lambda ^{2})\ xy-\lambda x^{2}=0$$ is


A
45o
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B
60o
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C
90o
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D
15o
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Solution

The correct option is C $$90^{o}$$
Pair of straight lines through origin.

$$ y = m_1 x, \, y = m_2 x$$

$$(y - m_1 x)(y - m_2 x) =0$$

$$y^2 - (m_1 + m_2) xy + m_1 m_2 x^2 = 0$$

comparing with $$\lambda y^2 + (1 - \lambda^2) xy - \lambda x^2 = 0$$
$$\dfrac{1}{\lambda} = -\dfrac{(m_1 + m_2)}{(1 - \lambda^2)} = \dfrac{m_1m_2}{-\lambda}$$

$$m_1 + m_2 = \dfrac{\lambda^2 - 1}{\lambda} \,\, m_1 m_2 = -1$$
lines are having $$90^o$$ angle between them.

C is correct

Maths

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