<p>A tangent to the ellipse \(16x^2 + 9y^2 = 144\) making equal intercepts on both the axes is</p>
Step-by-Step Solution
Key Concept: A line making equal intercepts on both axes has the form x + y = a or x - y = a. Substitute this into the ellipse equation and use the tangency condition (discriminant = 0) to find the intercept value.
<p><strong>Step 1:</strong> Rewrite the ellipse in standard form: $\frac{x^2}{9} + \frac{y^2}{16} = 1$</p><p><strong>Step 2:</strong> A line with equal intercepts has form $x + y = a$ or $x - y = a$ (the magnitudes of intercepts are equal)</p><p><strong>Step 3:</strong> For $x + y = a$, substitute $y = a - x$ into the ellipse: $16x^2 + 9(a-x)^2 = 144$</p><p>$16x^2 + 9(a^2 - 2ax + x^2) = 144$</p><p>$25x^2 - 18ax + 9a^2 - 144 = 0$</p><p><strong>Step 4:</strong> For tangency, discriminant = 0: $(18a)^2 - 4(25)(9a^2 - 144) = 0$</p><p>$324a^2 - 900a^2 + 14400 = 0$</p><p>$-576a^2 + 14400 = 0$</p><p>$a^2 = 25$, so $a = ±5$</p><p><strong>Step 5:</strong> Similarly for $x - y = a$, we get $a = ±5$</p><p><strong>Step 6:</strong> The four tangents are: $x + y = 5$, $x + y = -5$, $x - y = 5$, $x - y = -5$</p><p>∴ Answer: A</p>
Correct Answer: A