Question:

<p><strong>Step 1: Apply the right angle condition at center</strong></p><p>For a chord of the hyperbola $\frac{x^2}{16} - \frac{y^2}{18} = 1$ to subtend a right angle at the center O(0,0), if the chord meets the hyperbola at points P and Q, then $\vec{OP} \cdot \vec{OQ} = 0$.</p><p>For the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ where $a^2 = 16, b^2 = 18$, the condition for a chord $x\cos\alpha + y\sin\alpha = p$ to subtend a right angle at the center is:</p><p>$$\frac{1}{a^2} - \frac{1}{b^2} = \frac{1}{p^2}$$</p><p><strong>Step 2: Calculate using the hyperbola parameters</strong></p><p>Substituting $a^2 = 16$ and $b^2 = 18$:</p><p>$$\frac{1}{16} - \frac{1}{18} = \frac{1}{p^2}$$</p><p>$$\frac{18 - 16}{16 \times 18} = \frac{1}{p^2}$$</p><p>$$\frac{2}{288} = \frac{1}{p^2}$$</p><p>$$\frac{1}{144} = \frac{1}{p^2}$$</p><p>$$p^2 = 144$$</p><p>$$p = 12$$</p><p><strong>Step 3: Find the diameter of the concentric circle</strong></p><p>The chord $x\cos\alpha + y\sin\alpha = 12$ is tangent to a circle concentric with the hyperbola (center at origin). The perpendicular distance from the origin to this line equals the radius of the circle:</p><p>$$r = \frac{|0 \cdot \cos\alpha + 0 \cdot \sin\alpha - 12|}{\sqrt{\cos^2\alpha + \sin^2\alpha}} = \frac{12}{1} = 12$$</p><p><strong>Step 4: Calculate d/4</strong></p><p>The diameter of the circle is:</p><p>$$d = 2r = 2 \times 12 = 24$$</p><p>Therefore:</p><p>$$\frac{d}{4} = \frac{24}{4} = 6$$</p><p><strong>∴ Answer: C</strong></p>