- The midpoint of the horizontal chord \(PQ\) on the parabola \(y = x^2 - 10x + 20\) is calculated by finding the average \(x\)-coordinate where \(f(x_1) = f(x_2)\), resulting in \(M = (5, -5)\).
- The line of symmetry of the parabola is the vertical line passing through these midpoints, given by the equation \(\boxed{x=5}\).
Explanation
- Analysis and Midpoint Calculation
The given curve is a parabola defined by the quadratic equation \(y = x^2 - 10x + 20\).
A parabola of the form \(y = ax^2 + bx + c\) is symmetric about a vertical line passing through its vertex.
Points \(P\) and \(Q\) lying on the curve mean their coordinates satisfy this equation.
The question asks for the midpoint of the chord \(PQ\). In the general case of a parabola, the midpoint of any horizontal chord (a chord where \(P\) and \(Q\) share the same \(y\)-coordinate) lies on the axis of symmetry of the parabola.
Let \(P = (x_1, y_1)\) and \(Q = (x_2, y_1)\). Since they lie on the curve, we have:
\[y_1 = x_1^2 - 10x_1 + 20\]
\[y_1 = x_2^2 - 10x_2 + 20\]
Subtracting these gives \(x_1^2 - 10x_1 = x_2^2 - 10x_2\), which rearranges to \(x_1^2 - x_2^2 - 10(x_1 - x_2) = 0\).
Factoring, \((x_1 - x_2)(x_1 + x_2 - 10) = 0\).
Assuming \(x_1
eq x_2\), we have \(x_1 + x_2 = 10\).
The \(x\)-coordinate of the midpoint \(M\) is \(\frac{x_1 + x_2}{2} = \frac{10}{2} = 5\).
The \(y\)-coordinate of the midpoint is the value of the function at \(x=5\): \(y = 5^2 - 10(5) + 20 = 25 - 50 + 20 = -5\).
Thus, the midpoint of the chord \(PQ\) is \((5, -5)\).
- Determining the Line of Symmetry
The line of symmetry of a parabola \(y = ax^2 + bx + c\) is a vertical line that passes through the vertex and the midpoints of all horizontal chords.
We have determined that for any horizontal chord \(PQ\), the midpoint has an \(x\)-coordinate of \(5\).
Therefore, the line of symmetry is the vertical line passing through all such midpoints.
The equation of this vertical line is \(x = 5\).
Examples
In engineering, symmetry is crucial for designing balanced structures. If you are building a suspension bridge, the cables on one side must mirror the other to distribute weight evenly. By finding the axis of symmetry of a parabolic arch, architects ensure that the structure remains stable and aesthetically pleasing. Understanding this mathematical property helps in calculating central points of load for arches, which is essential for safety and efficiency in civil engineering.