Question:
<p>The line lx + my + n = 0 will be a tangent to the circle x<sup>2</sup> + y<sup>2</sup> = a<sup>2</sup> if</p>
<p style="display:inline">n<sup>2</sup>(l<sup>2</sup> + m<sup>2</sup>) = a<sup>2</sup></p>
<p style="display:inline">n(l + m) = a</p>
<p style="display:inline">a<sup>2</sup>(I<sup>2</sup> + m<sup>2</sup>) = n<sup>2</sup></p>
<p style="display:inline">a(l + m) = n</p>
Step-by-Step Solution
Key Concept: To determine tangency, convert the general line equation into slope-intercept form and substitute the resulting slope and intercept into the condition $c^2 = a^2(1+m^2)$.
<p>Line y = mx + c is a tangent, if c = <span class="math-tex">$\pm a \sqrt{1+m^{2}}$</span>.<br />
Now lx + my + n = 0 or <span class="math-tex">$y=-\frac{l}{m} x-\frac{n}{m}$</span> is a tangent, if<br />
<span class="math-tex">$-\frac{n}{m}=\pm a \sqrt{1+\left(\frac{l}{m}\right)^{2}}$</span><br />
i.e., if n<sup>2</sup> = a<sup>2</sup>(m<sup>2</sup> + l<sup>2</sup>)</p>
Correct Answer: C