Question:
<p>If p and q represent the lengths of a direct common tangent and a transverse common tangent respectively to the circles <span class="math-tex">\(\omega_1\)</span> : x<sup>2</sup> + y<sup>2</sup> + 14x - 4y + 28 = 0 and <span class="math-tex">\(\omega_2\)</span> : x<sup>2</sup> + y<sup>2</sup> - 14x + 4y - 28 = 0, then <span class="math-tex">\(\frac{p}{q}\)</span> is</p>
<p style="display:inline">7</p>
<p style="display:inline">4</p>
<p style="display:inline">14</p>
<p style="display:inline"><span class="math-tex">\(\frac{7}{2}\)</span></p>
Step-by-Step Solution
<html><body><p><span class="math-tex">$\omega_1$</span> : (x + 7)<sup>2</sup>+ (y - 2)<sup>2</sup> = 25<br/>
<span class="math-tex">$\Rightarrow$</span> Centre C<sub>1</sub> = (-7, 2) radius = 5<br/>
<span class="math-tex">$\omega_2$</span> : (x - 7)<sup>2</sup> + (y + 2)<sup>2</sup> = 81<br/>
<span class="math-tex">$\Rightarrow$</span> Centre C<sub>2</sub> = (7, -2), radius = 9<br/>
|C<sub>1</sub>C<sub>2</sub>| <span class="math-tex">$=\sqrt{(7+7)^{2}+(-2-2)^{2}}=\sqrt{212}$</span><br/>
<span class="math-tex">$\Rightarrow$</span> |C<sub>1</sub> C<sub>2</sub>| > sum of the radii<br/>
<span class="math-tex">$\Rightarrow$</span> Circles are apart.<br/>
<img alt="" data-imgur-src="ny8HYL0.png" src="https://media-mycbseguide.s3.amazonaws.com/images/imgur/1623761579-8vhsyv.jpg" style="width: 150px; height: 74px;"/><br/>
Length of transverse common tangent = |AB|<br/>
= |C<sub>1</sub>M|<br/>
<span class="math-tex">$=\sqrt{\left(\mathrm{C}_{1} \mathrm{C}_{2}\right)^{2}-16}$</span><br/>
<span class="math-tex">$=\sqrt{212-16}=\sqrt{196}$</span><br/>
<span class="math-tex">$\Rightarrow$</span> p = 14<br/>
<img alt="" data-imgur-src="o522DQY.png" src="https://media-mycbseguide.s3.amazonaws.com/images/imgur/1623761582-x9hf4r.jpg" style="width: 130px; height: 87px;"/><br/>
Length of transverse common tangent = |PN|<br/>
= |C<sub>2</sub>C<sub>1</sub>'|<br/>
<span class="math-tex">$=\sqrt{\left(\mathrm{C}_{1} \mathrm{C}_{2}\right)^{2}-(9+5)^{2}}$</span><br/>
<span class="math-tex">$=\sqrt{212-196}=\sqrt{16}$</span><br/>
<span class="math-tex">$\Rightarrow$</span> q = 4<br/>
<span class="math-tex">$\Rightarrow \frac{p}{q}=\frac{14}{4}=\frac{7}{2}$</span></p></body></html>
Correct Answer: D