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प्रश्न
If θ is the acute angle between the lines represented by ax2 + 2hxy + by2 = 0 then prove that tan θ = `|(2sqrt(h^2 - ab))/(a + b)|`
उत्तर
Proof: Let m1 and m2 be slopes of lines represented by the equation ax2 + 2hxy + by2 = 0
∴ m1 + m2 = `- (2h)/b` and m1m2 = `a/b`
∴ (m1 – m2)2 = (m1 + m2)2 – 4m1m2
= `(-(2h)/b)^2 - 4(a/b)`
= `(4h^2)/b^2 - (4ab)/b^2`
= `(4h^2 - 4ab)/b^2`
= `(4(h^2 - ab))/b^2`
∴ m1 – m2 = `± (2sqrt(h^2 - ab))/b`
As θ is the acute angle between the lines,
tan θ = `|(m_1 - m_2)/(1 + m_1m_2)|`
= `|(± (2sqrt(h^2 - ab))/b)/(1 + a/b)|`
= `|(2sqrt(h^2 - ab))/(a + b)|`
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