Circles
Tangent Lines to Circles
Grade 11
Question:
<p>A line has equation \(\frac{x}{a} + \frac{y}{b} = 1\). If P is the foot of the perpendicular drawn from the origin to this line, with coordinates \((x_1, y_1)\), and the distance from centre \((2, 4)\) to the line \(3x - 4y - k = 0\) equals the radius \(5\), find the value of \(k\).</p>
Step-by-Step Solution
Key Concept: The distance from a point to a line is given by the perpendicular distance formula, which must equal the radius for tangency.
<p><strong>Step 1:</strong> The distance from centre $(2, 4)$ to the line $3x - 4y - k = 0$ is:</p><p>$$d = \frac{|3(2) - 4(4) - k|}{\sqrt{9 + 16}} = \frac{|6 - 16 - k|}{5} = \frac{|-10 - k|}{5}$$</p><p><strong>Step 2:</strong> This distance must equal the radius $5$:</p><p>$$\frac{|-10 - k|}{5} = 5$$</p><p>$$|-10 - k| = 25$$</p><p><strong>Step 3:</strong> Solving: $-10 - k = \pm 25$</p><p>If $-10 - k = 25$, then $k = -35$</p><p>If $-10 - k = -25$, then $k = 15$</p><p><strong>Step 4:</strong> Since $k > 0$, we have $k = 15$.</p>
Correct Answer: 15