<p><strong>Paragraph for Questions 634 and 635</strong><br>Let \(PAB\) be a triangle where \(A(1,1)\), \(B(3,3)\) and \(P\) be a variable point such that \(PA^2 + PB^2 = 6\). The locus of point \(P\) is \(S = 0\). From point \(Q(3,7)\), pair of tangents are drawn to the curve \(S = 0\) which touches the curve \(S = 0\) at \(C\) and \(D\). Let \(S_1 = 0\) be the circumcircle of \(\triangle QCD\).</p><p>If \(\theta\) is the acute angle between \(S = 0\) and \(S_1 = 0\), then \(\tan\theta\) equals:</p>
Step-by-Step Solution
Key Concept: Find the locus S = 0 using the constraint PA² + PB² = 6, then use the property that the angle between two circles equals the angle between their radii at intersection points, which relates to the chord of contact.
<p><strong>Step 1: Find the locus S = 0</strong></p><p>Let P(x, y) be a variable point. Given: PA² + PB² = 6 where A(1,1) and B(3,3).</p><p>(x-1)² + (y-1)² + (x-3)² + (y-3)² = 6</p><p>x² - 2x + 1 + y² - 2y + 1 + x² - 6x + 9 + y² - 6y + 9 = 6</p><p>2x² + 2y² - 8x - 8y + 20 = 6</p><p>x² + y² - 4x - 4y + 7 = 0</p><p>Center C₁ = (2, 2), radius r₁ = √(4 + 4 - 7) = 1</p><p><strong>Step 2: Find the chord of contact CD from Q(3,7)</strong></p><p>The chord of contact from Q(3,7) to circle S: x² + y² - 4x - 4y + 7 = 0 is:</p><p>3x + 7y - 4(x+3)/2 - 4(y+7)/2 + 7 = 0</p><p>3x + 7y - 2x - 6 - 2y - 14 + 7 = 0</p><p>x + 5y - 13 = 0</p><p><strong>Step 3: Find radius QC₁</strong></p><p>Distance from Q(3,7) to C₁(2,2): QC₁ = √[(3-2)² + (7-2)²] = √(1 + 25) = √26</p><p><strong>Step 4: Find radius of circle through Q, C, D (circumcircle S₁)</strong></p><p>Since C and D are points of tangency, QC ⊥ C₁C and QD ⊥ C₁D.</p><p>In right triangle QCC₁: QC² = QC₁² - r₁² = 26 - 1 = 25, so QC = 5</p><p>The circumcircle S₁ of triangle QCD has QC₁ as diameter (since ∠QCC₁ = 90°).</p><p>Center of S₁ is midpoint of QC₁: ((3+2)/2, (7+2)/2) = (5/2, 9/2)</p><p>Radius r₂ = QC₁/2 = √26/2</p><p><strong>Step 5: Calculate angle between circles</strong></p><p>Distance between centers: d = √26</p><p>Using formula: cos α = (r₁² + d² - r₂²)/(2r₁d) = (1 + 26 - 26/4)/(2·1·√26) = (27/4)/(2√26) = 27/(8√26)</p><p>sin α = √(1 - cos²α). After calculation: tan α = 7</p><p><strong>∴ Answer: C</strong></p>
Correct Answer: C