Derive an expression for the angle between two lines y=m1x+c1 and y=m2x+c2
Answers
Answer:
tan⁻¹ (m₁ + m₂)/(1 - m₁m₂)
Step-by-step explanation:
In any equation in form of y = mx + c, m is the slope of the line i.e. tangent of the angle between +ve x-axis and that line.
So,
If A is the angle between x-axis and y = m₁x + c₁, then tanA = m₁, thus, tan⁻¹ m₁ = A
Similarly, if B is the angle between x-axis and y = m₂x + c₂, then tanB = m₂, thus, tan⁻¹ m₂= B
In the triangle formed by the joining lines y = m₁x + c₁ and y = m₂x + c₂, angles are A, C and π - B(B is made by 2nd line and x-axis, so π - B is the in that ∆).
=> A + (π - B) + C = 180°, note that either C or π - C is the angle between the lines.
A + B = C = angle between the lines.
tan⁻¹ m₁ + tan⁻¹ m₂ = C
tan⁻¹ (m₁ + m₂)/(1 - m₁m₂) = C
Hence the angle between two lines is tan⁻¹ | (m₁ + m₂)/(1 - m₁m₂) |, as it must be +ve. Moreover it can be π - tan⁻¹ | (m₁ + m₂)/(1 - m₁m₂) | as well.
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Answer:
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Step-by-step explanation:
Answer is 30 and 150
