<p>Let the intercepts be <em>a</em> and <em>b</em> such that <em>a</em> + <em>b</em> = −1. The line passes through the point (4, 3). How many equations of straight lines satisfy these conditions?</p>
Step-by-Step Solution
Key Concept: A line with intercepts a and b has equation x/a + y/b = 1. Use the constraint a + b = −1 and the point (4,3) to set up equations in a and b, then count distinct lines.
<p><strong>Step 1:</strong> Write the intercept form. A line with x-intercept a and y-intercept b is: x/a + y/b = 1</p><p><strong>Step 2:</strong> Apply constraint a + b = −1, so b = −1 − a</p><p><strong>Step 3:</strong> Substitute into intercept form: x/a + y/(−1−a) = 1</p><p><strong>Step 4:</strong> The line passes through (4, 3): 4/a + 3/(−1−a) = 1</p><p><strong>Step 5:</strong> Simplify: 4/a − 3/(1+a) = 1</p><p>Multiply through by a(1+a): 4(1+a) − 3a = a(1+a)</p><p>4 + 4a − 3a = a + a²</p><p>4 + a = a + a²</p><p>a² = 4</p><p>a = 2 or a = −2</p><p><strong>Step 6:</strong> For a = 2: b = −3. Line: x/2 + y/(−3) = 1 ✓ (Check: 4/2 + 3/(−3) = 2 − 1 = 1 ✓)</p><p>For a = −2: b = 1. Line: x/(−2) + y/1 = 1 ✓ (Check: 4/(−2) + 3/1 = −2 + 3 = 1 ✓)</p><p><strong>Step 7:</strong> Both give valid distinct lines. Also check if a line through (4,3) with slope m exists satisfying a + b = −1: Such a line y − 3 = m(x − 4) has intercepts a = 4m−3/m and b = 3−4m. Then a + b = −1 gives one more constraint, but this reduces to the quadratic already solved.</p><p>∴ Answer: <strong>2 equations</strong> (B)</p>
Correct Answer: B