<p>The lines 3<i>x</i> - 4<i>y</i> + 4 = 0 and 6<i>x</i> - 8<i>y</i> - 7 = 0 are tangents to the same circle whose radius is <i>r</i>, then 4<i>r</i> is equal to ?</p>
Step-by-Step Solution
Key Concept: For two parallel lines to be tangents to the same circle, the radius equals half the perpendicular distance between them. First verify the lines are parallel by checking their slopes are equal.
<p><strong>Step 1: Verify the lines are parallel.</strong></p><p>Line 1: 3x - 4y + 4 = 0 has slope m₁ = 3/4</p><p>Line 2: 6x - 8y - 7 = 0 can be rewritten as 6x - 8y - 7 = 0, which gives slope m₂ = 6/8 = 3/4</p><p>Since m₁ = m₂, the lines are parallel. ✓</p><p><strong>Step 2: Rewrite lines in standard form for distance calculation.</strong></p><p>Line 1: 3x - 4y + 4 = 0</p><p>Line 2: 6x - 8y - 7 = 0</p><p>To compare, make coefficients of x and y identical. Multiply Line 1 by 2:</p><p>Line 1': 6x - 8y + 8 = 0</p><p>Line 2: 6x - 8y - 7 = 0</p><p><strong>Step 3: Calculate perpendicular distance between parallel lines.</strong></p><p>For lines a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0 (with equal coefficients a and b):</p><p>Distance d = |c₁ - c₂|/√(a² + b²)</p><p>d = |8 - (-7)|/√(6² + (-8)²) = |8 + 7|/√(36 + 64) = 15/√100 = 15/10 = 3/2</p><p><strong>Step 4: Find the radius.</strong></p><p>Since both lines are tangents to the same circle on opposite sides, the distance between them equals the diameter:</p><p>Diameter = d = 3/2</p><p>Therefore: r = diameter/2 = (3/2)/2 = 3/4</p><p><strong>Step 5: Calculate 4r.</strong></p><p>4r = 4 × (3/4) = 3</p><p><strong>∴ Answer: 4r = 3</strong></p>
Correct Answer: 4