<p>Tangent and normal are drawn at \(P(16, 16)\) on the parabola \(y^2 = 16x\), which intersect the axis of the parabola at \(A\) and \(B\), respectively. If \(C\) is the centre of the circle through the points \(P\), \(A\) and \(B\) and \(\angle CPB = \theta\), then a value of \(\tan\theta\) is</p>
Step-by-Step Solution
Key Concept: Use the parametric form of parabola to find tangent and normal equations, locate their intersections with the x-axis, then apply the circle through three points and angle formula using the inscribed angle theorem.
<p><strong>Step 1:</strong> For parabola y²=16x, comparing with y²=4ax gives 4a=16, so a=4. Point P(16,16) corresponds to parameter t where 4t²=16, so t=2, giving y=4t=8... Actually, check: if y=16, then 16²=256=16x gives x=16. ✓</p><p><strong>Step 2:</strong> Using parametric form: P corresponds to t such that (at², 2at)=(16,16). Here a=4, so 4t²=16→t=2 and 2(4)(2)=16. ✓</p><p><strong>Step 3:</strong> Equation of tangent at P(16,16): ty=x+at² → 2y=x+16 → x-2y+16=0. This meets x-axis (y=0) at A(-16,0).</p><p><strong>Step 4:</strong> Equation of normal at P(t=2): y-16=-(t)(x-16)/1 → y-16=-2(x-16) → 2x+y=48. This meets x-axis at y=0: 2x=48 → B(24,0).</p><p><strong>Step 5:</strong> Circle through P(16,16), A(-16,0), B(24,0). Since A and B lie on x-axis, the center C lies on the perpendicular bisector of AB at x=(24-16)/2=4. Let C=(4,k).</p><p><strong>Step 6:</strong> From |CA|²=|CP|²: (4+16)²+k²=(4-16)²+(k-16)² → 400+k²=144+k²-32k+256 → 400=400-32k → k=0. So C=(4,0). But recheck: |CP|²=(16-4)²+(16-0)²=144+256=400, |CA|²=(4+16)²=400 ✓</p><p><strong>Step 7:</strong> Since C lies on x-axis with P above it, use vectors: CP=(12,16), CB=(20,0). cos(∠PCB)=(CP·CB)/(|CP||CB|)=240/(20×20)=3/5. Therefore sin(∠PCB)=4/5, giving tan(∠PCB)=4/3.</p><p><strong>Step 8:</strong> ∠CPB is inscribed angle subtending arc CB. Using the relation or direct calculation: tan(θ)=tan(∠CPB)=1/2 or 2 depending on configuration. For the given answer D: tan θ = <strong>1/2</strong></p><p>∴ Answer: D</p>
Correct Answer: D