Parabola
Tangent to Parabola
Grade 11
Question:
<p>If the line <span class="math">\frac{x}{l} + \frac{y}{m} = 1</span> touches the parabola <span class="math">y^2 = 4a(x + b)</span>, then <span class="math">m^2(l + b)</span> is equal to</p>
<p>(a) <span class="math">al^2</span></p>
<p>(b) <span class="math">-al^2</span></p>
<p>(c) <span class="math">l^2</span></p>
<p>(d) <span class="math">-a</span></p>
Step-by-Step Solution
Key Concept: Use coordinate transformation to shift the parabola to standard form, then apply the condition for tangency of a line to a parabola.
<p><strong>Step 1:</strong> The given parabola is <span class="math">y^2 = 4a(x + b)</span></p><p>Vertex of this parabola is <span class="math">(-b, 0)</span>.</p><p><strong>Step 2:</strong> Shift the origin to <span class="math">(-b, 0)</span>: Let <span class="math">x = X + (-b)</span> and <span class="math">y = Y</span></p><p>Then <span class="math">x + b = X</span> and <span class="math">y = Y</span></p><p>From the parabola equation: <span class="math">Y^2 = 4aX</span></p><p><strong>Step 3:</strong> The line <span class="math">\frac{x}{l} + \frac{y}{m} = 1</span> reduces to</p><p><span class="math">Y = -\frac{m}{l}X + m\left(1 + \frac{b}{l}\right)</span></p><p><strong>Step 4:</strong> For the line to touch the parabola <span class="math">Y^2 = 4aX</span>, apply the tangency condition:</p><p><span class="math">m\left(1 + \frac{b}{l}\right) = -\frac{a}{\frac{m}{l}}</span></p><p><span class="math">m^2\left(1 + \frac{b}{l}\right) = -al</span></p><p><span class="math">m^2\left(l + b\right) = -al^2</span></p><p>∴ Answer is (b) <span class="math">-al^2</span></p>
Correct Answer: B