<p>A ray of light passing through the point <span class="math inline">\(A(1, 2)\)</span> is reflected at a point <span class="math inline">\(B\)</span> on the x-axis and then passes through <span class="math inline">\((5, 3)\)</span>. Then the equation of AB is:</p>
<p>(a) <span class="math inline">\(5x + 4y = 13\)</span></p>
<p>(b) <span class="math inline">\(5x - 4y = -3\)</span></p>
<p>(c) <span class="math inline">\(4x + 5y = 14\)</span></p>
<p>(d) <span class="math inline">\(4x - 5y = -6\)</span></p>
Step-by-Step Solution
Key Concept: When light reflects off a surface, the angle of incidence equals the angle of reflection. This means the path taken is equivalent to a straight line from point A to the mirror image of point C(5,3) reflected across the x-axis.
<p><strong>Step 1:</strong> Identify the given information. Light passes through A(1, 2), reflects at point B on the x-axis, and then passes through C(5, 3).</p><p><strong>Step 2:</strong> Apply the law of reflection. Since B is on the x-axis (the mirror), the path from A to B to C is equivalent to a straight line from A(1, 2) to C'(5, -3), where C'(5, -3) is the reflection of C(5, 3) across the x-axis.</p><p><strong>Step 3:</strong> Find the equation of the line through A(1, 2) and C'(5, -3).</p><p>Slope = $\frac{-3 - 2}{5 - 1} = \frac{-5}{4}$</p><p><strong>Step 4:</strong> Using point-slope form with point A(1, 2):<br>$y - 2 = -\frac{5}{4}(x - 1)$<br>$4(y - 2) = -5(x - 1)$<br>$4y - 8 = -5x + 5$<br>$5x + 4y = 13$</p><p><strong>Step 5:</strong> Verify this line passes through A(1, 2): $5(1) + 4(2) = 5 + 8 = 13$ ✓</p><p><strong>∴ Answer:</strong> A</p>
Correct Answer: A