Question:
<p>If the tangents on the ellipse 4x<sup>2</sup> + y<sup>2</sup> = 8 at the points. (1, 2) and (a, b) are perpendicular to each other, then a<sup>2</sup> is equal to</p>
<p style="display:inline"><span class="math-tex">\(\frac{2}{17}\)</span></p>
<p style="display:inline"><span class="math-tex">\(\frac{128}{17}\)</span></p>
<p style="display:inline"><span class="math-tex">\(\frac{4}{17}\)</span></p>
<p style="display:inline"><span class="math-tex">\(\frac{64}{17}\)</span></p>
Step-by-Step Solution
Key Concept: Determine the slopes of the tangents at the given points using the T=0 formula and apply the perpendicularity condition m1m2 = -1.
<p>Equation of given ellipse is 4x<sup>2</sup> + y<sup>2</sup> = 8 ...(i)<br />
<span class="math-tex">\(\Rightarrow \quad \frac{x^{2}}{2}+\frac{y^{2}}{8}=1 \Rightarrow \frac{x^{2}}{(\sqrt{2})^{2}}+\frac{y^{2}}{(2 \sqrt{2})^{2}}=1\)</span><br />
Now, equation of tangent at point (1, 2) is 2x + y = 4 ...(ii)<br />
[<span class="math-tex">\(\because\)</span> equation of tangent to the ellipse <span class="math-tex">\(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\)</span> at (x<sub>1</sub>, x<sub>1</sub>) is <span class="math-tex">\(\frac{x x_{1}}{a^{2}}+\frac{y y_{1}}{b^{2}}=1\)</span>] and equation of another tangent at point (a, b)is<br />
4ax + by= 8 ... (iii)<br />
Since, lines (ii) and (iii) are perpendicular to each other.<br />
<span class="math-tex">\(\therefore\left(-\frac{2}{1}\right) \times\left(-\frac{4a}{b}\right)=-1\)</span><br />
[if lines a<sub>1</sub>x + b<sub>1</sub>y + c<sub>1</sub> = 0 and a<sub>2</sub>x + b<sub>2</sub>y + c<sub>2</sub> = 0 are perpendicular, then <span class="math-tex">\(\left(-\frac{a_{1}}{b_{1}}\right)\left(-\frac{a_{2}}{b_{2}}\right)=-1\)</span>]<br />
<span class="math-tex">\(\Rightarrow\)</span> b = -8a ...(iv)<br />
Also, the point (a, b) lies on the ellipse (i), so<br />
4a<sup>2</sup> + b<sup>2</sup> = 8<br />
4a<sup>2</sup> + 64a<sup>2</sup> = 8 [from Eq.(iv)]<br />
<span class="math-tex">\(\Rightarrow \quad 68 a^{2}=8 \Rightarrow a^{2}=\frac{8}{68}\)</span><br />
<span class="math-tex">\(\Rightarrow \quad a^{2}=\frac{2}{17}\)</span></p>
Correct Answer: A