Straight Lines
Perpendicular Lines
Grade None
Question:
<p>Given equation of straight line <em>2x</em> − 3<em>y</em> + 5 = 0 is perpendicular to the line passing through point (7, 17). If this perpendicular line also passes through point (15, β), find the value of β.</p>
Step-by-Step Solution
Key Concept: If two lines are perpendicular, the product of their slopes equals -1. Use this to find the slope of the perpendicular line, then apply the two-point form to find β.
<p><strong>Step 1:</strong> Find slope of given line 2x − 3y + 5 = 0</p><p>Rewrite as: 3y = 2x + 5 → y = (2/3)x + 5/3</p><p>Slope m₁ = 2/3</p><p><strong>Step 2:</strong> Find slope of perpendicular line</p><p>If m₁ · m₂ = -1, then: (2/3) · m₂ = -1</p><p>m₂ = -3/2</p><p><strong>Step 3:</strong> Use two-point form with points (7, 17) and (15, β)</p><p>Slope = (β − 17)/(15 − 7) = -3/2</p><p><strong>Step 4:</strong> Solve for β</p><p>(β − 17)/8 = -3/2</p><p>β − 17 = 8 × (-3/2) = -12</p><p>β = 17 − 12 = 5</p><p>∴ Answer: <strong>5</strong></p>
Correct Answer: 5