Ellipse
Tangent Parallel to a Line
Grade 11
Question:
<p>On the ellipse <i>4x</i><sup>2</sup> + <i>9y</i><sup>2</sup> = 1, the points at which the tangents are parallel to line 8<i>x</i> = 9<i>y</i> are:</p>
<p>(a) <span style='display:inline-block;text-align:center;'><span style='display:block;'>(</span><span style='display:block;'><span style='font-size:smaller;'>2</span>⁄<span style='font-size:smaller;'>5</span>, <span style='font-size:smaller;'>1</span>⁄<span style='font-size:smaller;'>5</span></span><span style='display:block;'>)</span></span></p>
<p>(b) <span style='display:inline-block;text-align:center;'><span style='display:block;'>(</span><span style='display:block;'>-<span style='font-size:smaller;'>2</span>⁄<span style='font-size:smaller;'>5</span>, <span style='font-size:smaller;'>1</span>⁄<span style='font-size:smaller;'>5</span></span><span style='display:block;'>)</span></span></p>
<p>(c) <span style='display:inline-block;text-align:center;'><span style='display:block;'>(</span><span style='display:block;'>-<span style='font-size:smaller;'>2</span>⁄<span style='font-size:smaller;'>5</span>, -<span style='font-size:smaller;'>1</span>⁄<span style='font-size:smaller;'>5</span></span><span style='display:block;'>)</span></span></p>
<p>(d) <span style='display:inline-block;text-align:center;'><span style='display:block;'>(</span><span style='display:block;'><span style='font-size:smaller;'>2</span>⁄<span style='font-size:smaller;'>5</span>, -<span style='font-size:smaller;'>1</span>⁄<span style='font-size:smaller;'>5</span></span><span style='display:block;'>)</span></span></p>
Step-by-Step Solution
Key Concept: Find points on the ellipse where the tangent line has the same slope as the given line. Use implicit differentiation or the tangent equation formula for ellipses to find where dy/dx equals the required slope.
<p><strong>Step 1: Find the slope of the given line.</strong></p><p>The line is 8x = 9y, which gives y = (8/9)x. So the required slope is m = 8/9.</p><p><strong>Step 2: Use implicit differentiation on the ellipse equation.</strong></p><p>Given: 4x² + 9y² = 1</p><p>Differentiating both sides with respect to x:</p><p>8x + 18y(dy/dx) = 0</p><p>dy/dx = -8x/(18y) = -4x/(9y)</p><p><strong>Step 3: Set the derivative equal to the required slope.</strong></p><p>We need: -4x/(9y) = 8/9</p><p>-4x/(9y) = 8/9</p><p>-4x = 8y</p><p>-x = 2y</p><p>x = -2y ... (equation I)</p><p><strong>Step 4: Substitute into the ellipse equation.</strong></p><p>From equation I: x = -2y</p><p>Substitute into 4x² + 9y² = 1:</p><p>4(-2y)² + 9y² = 1</p><p>4(4y²) + 9y² = 1</p><p>16y² + 9y² = 1</p><p>25y² = 1</p><p>y² = 1/25</p><p>y = ±1/5</p><p><strong>Step 5: Find corresponding x-coordinates.</strong></p><p>Using x = -2y:</p><p>When y = 1/5: x = -2(1/5) = -2/5 → Point: (-2/5, 1/5)</p><p>When y = -1/5: x = -2(-1/5) = 2/5 → Point: (2/5, -1/5)</p><p><strong>Step 6: Verify these points lie on the ellipse.</strong></p><p>For (-2/5, 1/5): 4(-2/5)² + 9(1/5)² = 4(4/25) + 9(1/25) = 16/25 + 9/25 = 25/25 = 1 ✓</p><p>For (2/5, -1/5): 4(2/5)² + 9(-1/5)² = 4(4/25) + 9(1/25) = 16/25 + 9/25 = 25/25 = 1 ✓</p><p><strong>∴ Answer: b, d</strong></p>
Correct Answer: b, d