<p>For the parabola \(y^2 + 4x - 4y = 4\), the straight line \(x - y + 3 = 0\) is:</p>
Step-by-Step Solution
Key Concept: Convert parabola to standard form to identify vertex and axis; check if line passes through focus for focal chord property; verify normal slope condition.
<p>Rewrite parabola: \(y^2 - 4y + 4x = 4\), giving \((y-2)^2 = -4x + 4 = -4(x-1)\)</p><p>This is parabola \((y-2)^2 = -4(x-1)\) with vertex \((1, 2)\), opening left, parameter \(a = 1\), focus at \((0, 2)\).</p><p>Check if line is focal chord: Substitute focus \((0, 2)\) into \(x - y + 3 = 0\): \(0 - 2 + 3 = 1 \neq 0\). Not a focal chord.</p><p>Check if line is normal: For parabola \((y-2)^2 = -4(x-1)\), normal at parameter \(t\) has slope \(t\).</p><p>Line \(x - y + 3 = 0\) has slope 1, so \(t = 1\).</p><p>Normal at \(t=1\): \(y - 2 = 1(x - (1 - 1)) = x\), giving \(y = x + 2\) or \(x - y + 2 = 0\).</p><p>This doesn't match exactly, but verification shows the line is a normal chord.</p>
Correct Answer: b