- Complete the square to rewrite the parabola as \(y = (x-5)^2 - 5\), identifying the vertex \((5, -5)\).
- Consider two points \(P\) and \(Q\) with the same y-coordinate; their x-coordinates satisfy the equation \(x^2 - 10x + (20 - k) = 0\).
- Calculate the midpoint of segment \(PQ\) using Vieta's formulas to find \(x_M = \frac{x_1 + x_2}{2} = 5\).
- Conclude that the midpoint is \((5, k)\) and the line of symmetry is \(\boxed{x = 5}\).
Explanation
- Analyzing the Quadratic Curve
The quadratic curve is given by \(y = x^2 - 10x + 20\). To understand its properties, we first complete the square for the expression:
\[y = (x^2 - 10x + 25) - 25 + 20 = (x - 5)^2 - 5\]
This vertex form tells us that the vertex of the parabola is at \((5, -5)\). The axis of symmetry for a parabola of the form \(y = a(x - h)^2 + k\) is the vertical line \(x = h\). Thus, our axis of symmetry is \(x = 5\).
- Calculating the Midpoint of Segment PQ
Let the coordinates of point \(P\) be \((x_1, y_1)\) and point \(Q\) be \((x_2, y_2)\). Since both points lie on the curve, they satisfy \(y_1 = x_1^2 - 10x_1 + 20\) and \(y_2 = x_2^2 - 10x_2 + 20\).
To find a meaningful midpoint that reflects the symmetry of the curve, we consider the case where \(P\) and \(Q\) have the same y-coordinate, \(y_1 = y_2 = k\).
Then, \(x^2 - 10x + 20 = k\), which rearranges to \(x^2 - 10x + (20 - k) = 0\).
By Vieta's formulas, the sum of the roots \(x_1 + x_2 = 10\).
The midpoint \(M(x_M, y_M)\) of segment \(PQ\) has coordinates:
\[x_M = \frac{x_1 + x_2}{2} = \frac{10}{2} = 5\]
\[y_M = \frac{y_1 + y_2}{2} = \frac{k + k}{2} = k\]
Thus, the midpoint is \((5, k)\), where \(k\) is the y-coordinate shared by points \(P\) and \(Q\).
- Determining the Line of Symmetry
From the calculation of the midpoint \(M = (5, k)\), we observe that the x-coordinate of the midpoint is constant at \(x = 5\), regardless of the choice of the shared y-coordinate \(k\).
Since the midpoint of a horizontal segment connecting two points on a parabola always lies on the axis of symmetry, and our x-coordinate of the midpoint is 5, we confirm that the vertical line \(x = 5\) is the axis of symmetry.
Therefore, the equation of the line of symmetry is \(x = 5\).
Examples
In architecture, the symmetry of a structure like a parabolic arch can be determined by finding the midpoint of horizontal support beams. If an architect knows that any two points at the same height on the arch have a midpoint with a fixed horizontal position, they can identify the central axis of the structure. This principle is applied in engineering to ensure the balance and structural integrity of symmetrical designs. For example, in the function \(y = x^2 - 10x + 20\), the midpoint of any horizontal chord will always lie on the vertical line \(x = 5\), which acts as the structural axis of symmetry.