<p>In each of the following one or more options are correct. Choose the correct option(s).</p><p>(a) The straight line <em>mx</em> − <em>y</em> + 1 + 2<em>m</em> cuts the circle <em>x</em>² + <em>y</em>² = 1 at one point at least. Then the set of values of <em>m</em> is</p>
<p>A. \(\left[-\frac{4}{3}, 0\right]\)</p>
<p>B. \(\left[-\frac{4}{3}, \frac{4}{3}\right]\)</p>
<p>C. \(\left[0, \frac{3}{4}\right]\)</p>
<p>D. none of these</p>
Step-by-Step Solution
Key Concept: Rewrite the line equation as m(x + 2) - y + 1 = 0 to recognize it passes through a fixed point (-2, 1). A line cuts a circle at least once if the distance from center to line is ≤ radius. Since (-2, 1) lies outside the unit circle, all lines through it will intersect the circle.
<p><strong>Step 1:</strong> Rewrite the line equation mx - y + 1 + 2m = 0 as m(x + 2) - (y - 1) = 0</p><p><strong>Step 2:</strong> This represents a family of lines passing through the fixed point P(-2, 1) for all values of m.</p><p><strong>Step 3:</strong> Check if P(-2, 1) lies inside, on, or outside the circle x² + y² = 1:</p><p>Distance from origin to P = √(4 + 1) = √5 > 1</p><p>So P lies outside the circle.</p><p><strong>Step 4:</strong> For a line passing through an external point, there are two cases: either it cuts the circle at 2 points or is tangent (1 point). It cannot miss the circle entirely.</p><p><strong>Step 5:</strong> Therefore, for ALL real values of m, the line cuts the circle at least once.</p><p>∴ Answer: m ∈ ℝ (all real numbers)</p>
Correct Answer: A