Question 10 of 70advanced🔧 ApplyShort Answer4 marks

The slope of a line is double of the slope of another line. If tangent of the angle between them is 13\frac{1}{3}, find the slopes of the lines.

Correct Answer

The slopes of the lines are: (i) 1/2 and 1, (ii) 1 and 2, (iii) -1/2 and -1, (iv) -1 and -2.

Exercise: EXERCISE 9.1 | Q: 10 | (Page 8)
For More Understanding

Explanation

This question applies the formula for the angle between two lines. The key insight is that when we remove the absolute value, we get two equations leading to two quadratic equations, each giving two values of m. The context shows the formula tan θ = |(m₂ - m₁)/(1 + m₁m₂)| which is essential for solving this problem.

Solution Steps

  1. Step 1: Let the slope of one line be m, then slope of other line is 2m.

  2. Step 2: Using the formula tan θ = |(m₂ - m₁)/(1 + m₁m₂)|, we have 1/3 = |(2m - m)/(1 + m·2m)|

  3. Step 3: Simplifying: 1/3 = |m/(1 + 2m²)|

  4. Step 4: This gives two equations: m/(1 + 2m²) = 1/3 OR m/(1 + 2m²) = -1/3

  5. Step 5: For m/(1 + 2m²) = 1/3: Cross-multiplying gives 3m = 1 + 2m², which gives 2m² - 3m + 1 = 0

  6. Step 6: Solving: (2m - 1)(m - 1) = 0, so m = 1/2 or m = 1

  7. Step 7: For m/(1 + 2m²) = -1/3: Cross-multiplying gives -3m = 1 + 2m², which gives 2m² + 3m + 1 = 0

  8. Step 8: Solving: (2m + 1)(m + 1) = 0, so m = -1/2 or m = -1

  9. Step 9: Therefore, the pairs of slopes are: (1/2, 1), (1, 2), (-1/2, -1), and (-1, -2).

PreviousNext