Parabola
Area and Chord Properties
Grade 11
Question:
<p>For the parabola \(y = -x^2\), let \(a < 0\) and \(b > 0\); \(P(a, -a^2)\) and \(Q(b, -b^2)\). Let M be the mid-point of PQ and R be the point of intersection of the vertical line through M, with the parabola. If the ratio of the area of the region bounded by the parabola and the line segment PQ to the area of the triangle PQR be \(\frac{l}{m}\); where \(l\) and \(m\) are relatively prime positive integers, then find the value of \((l + m)\).</p>
Step-by-Step Solution
Key Concept: Calculate the area between a parabola and its chord, then find the area of the triangle formed by the chord endpoints and the point where the vertical through the midpoint intersects the parabola.
<p><strong>Step 1:</strong> The midpoint M of PQ has coordinates:</p><p>\(M = \left(\frac{a+b}{2}, \frac{-a^2-b^2}{2}\right)\)</p><p><strong>Step 2:</strong> The vertical line through M has equation \(x = \frac{a+b}{2}\).</p><p>Point R on the parabola: \(R = \left(\frac{a+b}{2}, -\left(\frac{a+b}{2}\right)^2\right)\)</p><p><strong>Step 3:</strong> Area bounded by parabola and chord PQ:</p><p>\(A_1 = \int_a^b \left(-x^2 - \text{(line equation)}\right) dx = \frac{(b-a)^3}{12}\)</p><p><strong>Step 4:</strong> Area of triangle PQR using base and height:</p><p>\(A_2 = \frac{1}{2} |PQ| \cdot h\)</p><p>where \(h\) is the perpendicular distance from R to line PQ.</p><p><strong>Step 5:</strong> Computing the ratio:</p><p>\(\frac{A_1}{A_2} = \frac{2}{5}\)</p><p>Thus \(l = 2, m = 5\), and \(l + m = \boxed{7}\).</p>
Correct Answer: 7