Question 57 of 70intermediate🔧 ApplyShort Answer4 marks

Find the equation of the lines through the point (3,2)(3, 2) which make an angle of 45∘45^\circ with the line x−2y=3x - 2y = 3.

Correct Answer

The equations of the required lines are 3x - y - 7 = 0 and x + 3y - 9 = 0.

Exercise: Miscellaneous Exercise on Chapter 9 | Q: 10 | (Page 22)
For More Understanding

Explanation

This question tests the concept of angle between two lines using the formula tan θ = |(m₂ - m₁)/(1 + m₁m₂)|. The student needs to find the slope of the given line first, then use the angle formula to find two possible slopes for the required lines, and finally apply the point-slope form to get the equations.

Solution Steps

  1. Step 1: Find the slope of the given line x - 2y = 3. Rearranging: 2y = x - 3, so y = (1/2)x - 3/2. Therefore, slope m₁ = 1/2.

  2. Step 2: Let m be the slope of the required line through (3, 2). Using the angle formula: tan 45° = |(m - m₁)/(1 + m·m₁)| = 1

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

  4. Step 4: Case 1: m - 1/2 = 1 + m/2, solving: m/2 = 3/2, so m = 3. Using point-slope form: y - 2 = 3(x - 3), giving 3x - y - 7 = 0

  5. Step 5: Case 2: m - 1/2 = -1 - m/2, solving: 3m/2 = -1/2, so m = -1/3. Using point-slope form: y - 2 = (-1/3)(x - 3), giving x + 3y - 9 = 0

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