<p>If the equation of tangent to the circle <i>x</i><sup>2</sup> + <i>y</i><sup>2</sup> - 2<i>x</i> + 6<i>y</i> - 6 = 0 and parallel to 3<i>x</i> - 4<i>y</i> + 7 = 0 is 3<i>x</i> - 4<i>y</i> + <i>k</i> = 0, then the value of <i>k</i> are</p>
Step-by-Step Solution
Key Concept: For a line to be tangent to a circle, the perpendicular distance from the centre to the line must equal the radius. Since two parallel tangents exist, we get two values of k.
<p><strong>Solution:</strong> The centre and radius of given circle are (1, -3) and 4, respectively.</p><p>The length of perpendicular from centre (1, -3) to 3<i>x</i> - 4<i>y</i> + <i>k</i> = 0 is equal to radius 4.</p><p>$\frac{|3(1) - 4(-3) + k|}{\sqrt{9 + 16}} = 4$</p><p>$\frac{|3 + 12 + k|}{5} = 4$</p><p>$|15 + k| = 20$</p><p>$15 + k = \pm 20$</p><p>$k = 5 \text{ or } k = -35$</p><p>∴ Answer is (a).</p>
Correct Answer: A