Question 41 of 70advanced🔧 ApplyShort Answer4 marks

Two lines passing through the point (2,3)(2, 3) intersects each other at an angle of 60∘60^\circ. If slope of one line is 2, find equation of the other line.

Correct Answer

The equations of the other line are:

Equation 1: y - 3 = [(53\sqrt{3} - 8)/11](x - 2)

or equivalently: (8 - 53\sqrt{3})x + 11y - 49 + 103\sqrt{3} = 0

Equation 2: y - 3 = [-(8 + 53\sqrt{3})/11](x - 2)

or equivalently: (8 + 53\sqrt{3})x + 11y - 49 - 103\sqrt{3} = 0

Exercise: EXERCISE 9.3 | Q: 11 | (Page 17)
For More Understanding

Explanation

This problem requires finding the equation of a line passing through point (2, 3) that makes an angle of 60° with a line having slope 2. Using the angle formula between two lines and the point-slope form from the textbook, we find two possible lines since the angle can be measured in either direction.

Solution Steps

  1. Step 1: Let m₁ = 2 (slope of given line) and m₂ be the slope of the required line passing through (2, 3).

  2. Step 2: Using the angle formula: tan θ = |(m₁ - m₂)/(1 + m₁m₂)|, where θ = 60° and tan 60° = 3\sqrt{3}.

  3. Step 3: Substituting values: 3\sqrt{3} = |(2 - m₂)/(1 + 2m₂)|

  4. Step 4: This gives two cases: (2 - m₂)/(1 + 2m₂) = 3\sqrt{3} OR (2 - m₂)/(1 + 2m₂) = -3\sqrt{3}

  5. Step 5: Case 1: Solving (2 - m₂)/(1 + 2m₂) = 3\sqrt{3} gives m₂ = (53\sqrt{3} - 8)/11

  6. Step 6: Case 2: Solving (2 - m₂)/(1 + 2m₂) = -3\sqrt{3} gives m₂ = -(8 + 53\sqrt{3})/11

  7. Step 7: Using point-slope form y - y₀ = m(x - x₀) with point (2, 3):

  8. Step 8: For m₂ = (53\sqrt{3} - 8)/11: y - 3 = [(53\sqrt{3} - 8)/11](x - 2)

  9. Step 9: For m₂ = -(8 + 53\sqrt{3})/11: y - 3 = [-(8 + 53\sqrt{3})/11](x - 2)

PreviousNext