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The slopes of two straight lines are 1 and √3. What is the acute angle between these two lines?

Difficulty: Medium

Correct Answer: 15°

Explanation:


Introduction / Context:
This coordinate geometry question tests your understanding of the relationship between slopes of lines and the angle between them. When you know the slopes of two lines, you can find the angle between them using a standard formula involving the tangent of the angle. Here, the slopes are 1 and √3, which are chosen so that the final angle corresponds to a familiar special angle.


Given Data / Assumptions:

  • Slope of the first line m1 = 1.
  • Slope of the second line m2 = √3.
  • We are asked for the acute angle between the two lines.


Concept / Approach:
The formula for the tangent of the angle θ between two lines with slopes m1 and m2 is: tan θ = |(m2 - m1) / (1 + m1 * m2)|. We compute this ratio, simplify it, and then identify θ by recognizing the standard angle whose tangent equals the computed value. Finally, we choose the acute angle from the options.


Step-by-Step Solution:

Step 1: Use the formula tan θ = |(m2 - m1) / (1 + m1 * m2)|. Step 2: Substitute m1 = 1 and m2 = √3. Step 3: Compute the numerator: m2 - m1 = √3 - 1. Step 4: Compute the denominator: 1 + m1 * m2 = 1 + (1)(√3) = 1 + √3. Step 5: So tan θ = |(√3 - 1) / (1 + √3)|. Step 6: Multiply numerator and denominator by (1 - √3) to simplify: this is a standard rationalisation technique. Step 7: The denominator becomes (1 + √3)(1 - √3) = 1 - 3 = -2, and the numerator becomes (√3 - 1)(1 - √3) = -(√3 - 1)^2. Step 8: Therefore tan θ = |-(√3 - 1)^2 / -2| = (√3 - 1)^2 / 2. Step 9: Evaluate (√3 - 1)^2 = 3 + 1 - 2√3 = 4 - 2√3. Then (4 - 2√3)/2 = 2 - √3. Step 10: So tan θ = 2 - √3, which is the known exact value of tan 15°. Step 11: Hence θ = 15°, which is acute and matches one of the options.


Verification / Alternative check:
Using a calculator, approximate the value: √3 is about 1.732, so m2 - m1 ≈ 0.732 and 1 + m1 * m2 ≈ 2.732. The ratio is about 0.268. The tangent of 15 degrees is approximately 0.268, confirming that the acute angle between the lines is 15 degrees.


Why Other Options Are Wrong:

30°, 45°, 60°, 75°: each has a well known tangent value (√3/3, 1, √3, and about 3.732 respectively) that does not match the computed tan θ value of 2 - √3.


Common Pitfalls:
Students sometimes forget the absolute value in the formula or invert the fraction incorrectly. Another mistake is to misremember the formula as (m1 - m2)/(m1 + m2). However, the correct form uses 1 + m1 * m2 in the denominator. Being careful with the formula and using known tangent values of special angles helps avoid such errors.


Final Answer:
15°

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