Question:
<p>If the tangent at (1, 7) to the curve x<sup>2</sup> = y - 6 touches the circle x<sup>2</sup> + y<sup>2</sup> + 16x + 12y + c = 0, then the value of c is</p>
<p style="display:inline">195</p>
<p style="display:inline">185</p>
<p style="display:inline">95</p>
<p style="display:inline">85</p>
Step-by-Step Solution
Key Concept: Determine the tangent to the parabola at the given point and apply the condition that the perpendicular distance from the circle's center to this line equals the circle's radius.
<p>Tangent to the curve x<sup>2</sup> = y - 6 at (1, 7) is<br />
x = <span class="math-tex">$\frac{y+7}{2}$</span> - 6<br />
<span class="math-tex">$\Rightarrow$</span> 2x - y + 5 = 0 ......(i)<br />
Equation of circle is x<sup>2</sup> + y<sup>2</sup> + 16x + 12y + c = 0<br />
Centre (-8, -6)<br />
r = <span class="math-tex">$\sqrt{8^{2}+6^{2}-c}$</span> = <span class="math-tex">$\sqrt{100-c}$</span><br />
Since, line 2x - y + 5 = 0 also touches the circle.<br />
<span class="math-tex">$\therefore$</span> <span class="math-tex">$\sqrt{100-c}$</span> = <span class="math-tex">$\left|\frac{2(-8)-(-6)+5}{\sqrt{2^{2}+1^{2}}}\right|$</span><br />
<span class="math-tex">$\Rightarrow$</span> <span class="math-tex">$\sqrt{100-c}$</span> = <span class="math-tex">$\left|\frac{-16+6+5}{\sqrt{5}}{}{}\right|$</span><br />
<span class="math-tex">$\Rightarrow$</span> <span class="math-tex">$\sqrt{100-c}$</span> = <span class="math-tex">$|-\sqrt{5}|$</span><br />
<span class="math-tex">$\Rightarrow$</span> 100 - c = 5<br />
<span class="math-tex">$\Rightarrow$</span> c = 95</p>
Correct Answer: C