Question:
<p>If the common tangent to the parabolas, y<sup>2</sup> = 4x and x<sup>2</sup> = 4y also touches the circle, x<sup>2</sup> + y<sup>2</sup> = c<sup>2</sup>, then c is equal to:</p>
<p style="display:inline"><span class="math-tex">\(\frac{1}{2}\)</span></p>
<p style="display:inline"><span class="math-tex">\(\frac{1}{\sqrt{2}}\)</span></p>
<p style="display:inline"><span class="math-tex">\(\frac{1}{4}\)</span></p>
<p style="display:inline"><span class="math-tex">\(\frac{1}{2 \sqrt{2}}\)</span></p>
Step-by-Step Solution
Key Concept: Find the common tangent by equating the slope-form tangent expressions for both parabolas, then apply the condition that the circle's radius equals the perpendicular distance from its center to that line.
<p>Equation tangent to parabola y<sup>2</sup> = 4x with slope m be:<br />
<span class="math-tex">$y=m x+\frac{1}{m}$</span> ...(i)<br />
<span class="math-tex">$\because$</span> Equation of tangent to x<sup>2</sup> = 4y with slope m be :<br />
y = mx - am<sup>2</sup> ...(ii)<br />
From eq. (i) and (ii),<br />
<span class="math-tex">$\frac{1}{m}=-m^{2}$</span> <span class="math-tex">$\Rightarrow$</span> m = -1<br />
<span class="math-tex">$\therefore$</span> Equation tangent :x + y + 1 = 0<br />
It is tangent to circle x<sup>2</sup> + y<sup>2</sup> = c<sup>2</sup><br />
<span class="math-tex">$\Rightarrow c=\frac{1}{\sqrt{2}}$</span></p>
Correct Answer: B