Straight Lines
Angle Bisectors between Two Lines
Grade 11
Question:
<p>The equation of line bisecting the obtuse angle between <i>y</i> − <i>x</i> = 2 and 2<i>y</i> + <i>x</i> = 5 is</p><p>$$\frac{y - x - 2}{2} = \frac{x + 2y - 5}{n}$$</p><p>where <i>n</i> is</p>
<p>(a) 5</p>
<p>(b) 2</p>
<p>(c) 5</p>
<p>(d) None of the above</p>
Step-by-Step Solution
Key Concept: When the dot product of normal vectors is positive, the bisector with the positive sign gives the obtuse angle bisector. Identify the coefficients and apply the angle bisector formula.
<p><strong>Step 1:</strong> Rewrite the equations of given lines in standard form:</p><p>Line 1: <i>−x</i> + <i>y</i> = 2 ... (i)</p><p>Line 2: <i>x</i> + 2<i>y</i> = 5 ... (ii)</p><p><strong>Step 2:</strong> Check the sign of <i>a</i><sub>1</sub><i>a</i><sub>2</sub> + <i>b</i><sub>1</sub><i>b</i><sub>2</sub>:</p><p><i>a</i><sub>1</sub><i>a</i><sub>2</sub> + <i>b</i><sub>1</sub><i>b</i><sub>2</sub> = (−1)(1) + (1)(2) = −1 + 2 = 1 > 0</p><p><strong>Step 3:</strong> Since the product is positive, the equation of the bisector of the obtuse angle is given by the positive sign:</p><p>$$\frac{y - x - 2}{2} = \frac{x + 2y - 5}{5}$$</p><p><strong>Step 4:</strong> Therefore, <i>n</i> = 5.</p><p>∴ Answer is C.</p>
Correct Answer: C