Question:
<p>If y = mx + 4 is a tangent to both the parabolas, y<sup>2</sup> = 4x and x<sup>2</sup> = 2by, then b is equal to:</p>
<p style="display:inline">128</p>
<p style="display:inline">-64</p>
<p style="display:inline">-32</p>
<p style="display:inline">-128</p>
Step-by-Step Solution
Key Concept: Identify the slope by comparing the line to the standard tangent form of the first parabola, then apply the tangency condition D=0 to the intersection equation of the second parabola.
<p>y = mx + 4 ...(i)<br />
Tangent of y<sup>2</sup> = 4x is<br />
<span class="math-tex">$\Rightarrow y=m x+\frac{1}{m}$</span> ...(ii) [<span class="math-tex">$\because$</span> Equation of tangent of y<sup>2</sup> = 4 ax is <span class="math-tex">$y=m x+\frac{a}{m}$</span>]<br />
From (i) and (ii)<br />
<span class="math-tex">$4=\frac{1}{m} \Rightarrow m=\frac{1}{4}$</span><br />
So, the line <span class="math-tex">$y=\frac{1}{4} x+4$</span> is also tangent to the parabola<br />
x<sup>2</sup> = 2by, so solve both equations.<br />
<span class="math-tex">$x^{2}=2 b\left(\frac{x+16}{4}\right)$</span><br />
<span class="math-tex">$\Rightarrow$</span> 2x<sup>2</sup> - bx - 16b = 0<br />
<span class="math-tex">$\Rightarrow$</span> D = 0 [For tangent]<br />
<span class="math-tex">$\Rightarrow$</span> b<sup>2</sup> - 4 <span class="math-tex">$\times$</span> 2 <span class="math-tex">$\times$</span> (-16b) = 0<br />
<span class="math-tex">$\Rightarrow$</span> b<sup>2</sup> + 32 <span class="math-tex">$\times$</span> 4b = 0<br />
b = -128, b = 0 (not possible)</p>
Correct Answer: D