Circles
Intersection of Line and Circle
Grade 11
Question:
<p>The circle <i>x</i><sup>2</sup> + <i>y</i><sup>2</sup> = 4<i>x</i> + 8<i>y</i> = 5 intersects the line 3<i>x</i> - 4<i>y</i> = <i>m</i> at two distinct points. Find the range of <i>m</i>.</p>
<p>(a) -85 < <i>m</i> < -35</p>
<p>(b) -35 < <i>m</i> < 15</p>
<p>(c) 15 < <i>m</i> < 65</p>
<p>(d) 35 < <i>m</i> < 85</p>
Step-by-Step Solution
Key Concept: For a line to intersect a circle at two distinct points, the perpendicular distance from the centre to the line must be less than the radius.
<p><strong>Solution:</strong> Given, centre of circle is (2, 4) and radius is 5.</p><p>The line will intersect the circle at two distinct points if the distance of (2, 4) from 3<i>x</i> - 4<i>y</i> = <i>m</i> is less than the radius of the circle.</p><p>Distance = $\frac{|6 - 16 - m|}{5} = \frac{|-10 - m|}{5}$</p><p>We need: $\frac{|10 + m|}{5} < 5$</p><p>$|10 + m| < 25$</p><p>$-25 < 10 + m < 25$</p><p>$-35 < m < 15$</p><p>∴ Answer is (b).</p>
Correct Answer: B