Next level of pair of straight lines

Geometry Level 5

If a x 3 + b y 3 + c x 2 y + d x y 2 = 0 ax^{3}+by^{3}+cx^{2}y+dxy^{2} = 0 represents the three distinct straight lines, such that each line bisects the angle between the other two, then which of the following is correct?

d 2 > 5 b c d^{2}> 5bc d 2 < 5 b c d^{2} < 5bc a + 4 d = 0 a + 4d = 0 3 b + c = 0 3b + c = 0

Archit Tripathi by Archit Tripathi , 23, India

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1 solution

Archit Tripathi
Sep 16, 2016

This can happen if three lines are real and distinct as well as angle between any two adjacent sides is 2 π 3 \frac{2\pi}{3}
⇒ \Rightarrow f ( m ) f(m) = b m 3 bm^{3} + d m 2 dm^{2} + c m cm + a = 0 a = 0 has 3 3 distinct real roots and

m 1 − m 2 1 + m 1 m 2 \frac{m_1- m_2}{1+ m_1m_2} = m 2 − m 3 1 + m 2 m 3 \frac{m_2 - m_3}{1 + m_2m_3} = m 3 − m 1 1 + m 3 m 1 \frac{m_3 - m_1}{1 + m_3m_1} = ± 3 \pm\sqrt{3}

⇒ \Rightarrow 3 + m 1 m 2 + m 2 m 3 + m 3 m 1 = 0 3 + m_1m_2 + m_2m_3 + m_3m_1 = 0 and

3 b m 2 + 2 d m + c = 0 3bm^{2} + 2dm + c = 0 has roots α , β \alpha,\beta then f ( α ) f ( β ) < 0 f(\alpha)f(\beta) < 0 ⇒ \Rightarrow

3 b + c = 0 3b + c = 0

2 Helpful 0 Interesting 0 Brilliant 0 Confused

@Archit Tripathi Please explain the last bit of your solution.!

Ankit Kumar Jain - 3 years, 6 months ago

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