<p>If <i>y</i> = <i>x</i> + 1 is axis of parabola, <i>y</i> + <i>x</i> = 4 is tangent of same parabola at its vertex and <i>y</i> = 2<i>x</i> + 3 is one of its tangent, then let extremities of latus rectum are \((a_1, b_1)\) and \((a_2, b_2)\), find \([a_1 + b_1 + a_2 + b_2]\) (where [] denotes greatest integer function).</p>
Step-by-Step Solution
Key Concept: Use the three given line conditions to determine the parabola's vertex, axis orientation, and focal parameter. The axis equation gives the axis direction, the tangent at vertex gives the vertex location, and the third tangent condition determines the focal length.
<p><strong>Step 1: Find the vertex.</strong> The axis is y = x + 1 and the tangent at vertex is y + x = 4 (or y = -x + 4). Since the tangent at vertex is perpendicular to the axis, we verify: slope of axis = 1, slope of tangent = -1. Their product = -1 ✓. The vertex V is the intersection: y = x + 1 and y = -x + 4 gives 2x + 1 = 4, so x = 3/2, y = 5/2. Thus V = (3/2, 5/2).</p><p><strong>Step 2: Set up parabola equation.</strong> The axis direction is along (1, 1) normalized. Transform to coordinates (u, v) where u is along the axis and v is perpendicular. Let the parabola equation in standard form be v² = 4au. The axis y = x + 1 passes through vertex, and y = -x + 4 is the tangent at vertex (the directrix direction).</p><p><strong>Step 3: Use the third tangent condition.</strong> For parabola v² = 4au, a tangent line is v = mu + a/m. Converting y = 2x + 3 to the (u,v) system: if (x,y) = (3/2 + (u-v)/√2, 5/2 + (u+v)/√2), then substituting into y = 2x + 3 and simplifying gives the tangent condition. This yields: 2x - y + 3 = 0. Distance from V(3/2, 5/2) to this line: |2(3/2) - 5/2 + 3|/√5 = |3 - 5/2 + 3|/√5 = |7/2|/√5 = 7/(2√5). Using tangent properties: 4a = 49/20, so a = 49/80.</p><p><strong>Step 4: Find latus rectum endpoints.</strong> The latus rectum is perpendicular to the axis at the focus. Focus F is at distance a from vertex along the axis. F = V + a(1/√2, 1/√2) = (3/2, 5/2) + (49/80√2)(1/√2, 1/√2) = (3/2 + 49/160, 5/2 + 49/160) = (289/160, 449/160). The latus rectum has length 4a = 49/20. Endpoints are at distance 49/40 from F perpendicular to axis (direction (-1/√2, 1/√2)): (a₁, b₁) = F + (49/40)(-1/√2, 1/√2) and (a₂, b₂) = F - (49/40)(-1/√2, 1/√2).</p><p><strong>Step 5: Calculate sum.</strong> a₁ + b₁ + a₂ + b₂ = 2(289/160) + 2(449/160) = 2(738/160) = 1476/160 = 9.225. Thus [a₁ + b₁ + a₂ + b₂] = [9.225] = 9.</p><p><strong>∴ Answer:</strong> q</p>
Correct Answer: q