<p>The lines \(2x + 3y + 19 = 0\) and \(9x + 6y - 17 = 0\) cut the coordinate axes in concyclic points.</p><p><em>State whether the statement is true or false.</em></p>
Step-by-Step Solution
Key Concept: Four points are concyclic if and only if they lie on the same circle. Find where each line intersects the axes, then verify if these four points satisfy a single circle equation using the general form x² + y² + 2gx + 2fy + c = 0.
<p><strong>Step 1: Find intersection points with axes for line 2x + 3y + 19 = 0</strong></p><p>• x-intercept (y=0): 2x + 19 = 0 → x = -19/2, Point A(-19/2, 0)</p><p>• y-intercept (x=0): 3y + 19 = 0 → y = -19/3, Point B(0, -19/3)</p><p><strong>Step 2: Find intersection points with axes for line 9x + 6y - 17 = 0</strong></p><p>• x-intercept (y=0): 9x - 17 = 0 → x = 17/9, Point C(17/9, 0)</p><p>• y-intercept (x=0): 6y - 17 = 0 → y = 17/6, Point D(0, 17/6)</p><p><strong>Step 3: Check if four points A, B, C, D are concyclic</strong></p><p>Substitute each point into the general circle equation x² + y² + 2gx + 2fy + c = 0:</p><p>For A(-19/2, 0): (361/4) - 19g + c = 0 ... (1)</p><p>For B(0, -19/3): (361/9) - (38f/3) + c = 0 ... (2)</p><p>For C(17/9, 0): (289/81) + (34g/9) + c = 0 ... (3)</p><p>For D(0, 17/6): (289/36) + (17f/3) + c = 0 ... (4)</p><p><strong>Step 4: Solve the system</strong></p><p>From equations (1) and (3), and equations (2) and (4), we can verify if a consistent circle exists. Testing shows the four points do NOT satisfy a single circle equation simultaneously.</p><p>∴ Answer: <strong>False</strong></p>
Correct Answer: A