- Calculate the midpoint of the segment \(PQ\) using the coordinates of \(P(1, 3)\) and \(Q(7, 3)\), yielding \(M(4, 3)\).
- Recognize that the midpoint of any horizontal chord on a parabola lies on its axis of symmetry.
- Define the line of symmetry as the vertical line passing through the x-coordinate of the midpoint.
- Confirm the axis of symmetry is \(x = 4\).
- Final results: Midpoint: \((4, 3)\); Line of symmetry: \(x = 4\).
Explanation
- Calculating the Midpoint of the Segment PQ
The curve is given by the equation \(y = x^2 - 8x + 10\). We are given two points on this curve, \(P(1, 3)\) and \(Q(7, 3)\). These points share the same y-coordinate, meaning they form a horizontal chord on the parabola. To find the midpoint of the line segment \(PQ\), we use the midpoint formula:
\[M = \left(\frac{x_P + x_Q}{2}, \frac{y_P + y_Q}{2}\right)\]
Substituting the coordinates of \(P(1, 3)\) and \(Q(7, 3)\):
\[M = \left(\frac{1 + 7}{2}, \frac{3 + 3}{2}\right)\]
\[M = \left(\frac{8}{2}, \frac{6}{2}\right) = (4, 3)\]
Thus, the coordinates of the midpoint of \(PQ\) are \((4, 3)\).
- Determining the Line of Symmetry
A fundamental property of a parabola is that it is symmetric about a vertical line known as the axis of symmetry. For any horizontal chord connecting two points on the parabola, the midpoint of that chord will always lie on the axis of symmetry.
Since the points \(P(1, 3)\) and \(Q(7, 3)\) have the same y-coordinate, the segment \(PQ\) is a horizontal chord. We have already determined that its midpoint is \(M(4, 3)\). Because this point must lie on the axis of symmetry, and the axis of symmetry for a parabola is a vertical line of the form \(x = h\), the axis of symmetry must pass through the x-coordinate of the midpoint.
Therefore, the equation of the line of symmetry is \(x = 4\). We can verify this by using the standard form of the axis of symmetry for a parabola \(y = ax^2 + bx + c\), which is \(x = -\frac{b}{2a}\). For \(y = x^2 - 8x + 10\), we have \(a = 1\) and \(b = -8\):
\[x = -\frac{-8}{2(1)} = \frac{8}{2} = 4\]
This confirms our result. The final answers are the midpoint \((4, 3)\) and the equation of the line of symmetry \(x = 4\).
Examples
Understanding the symmetry of a parabola is useful in engineering and design, such as when constructing a satellite dish or a bridge arch. By identifying the line of symmetry, engineers can ensure that the structure is perfectly balanced and that forces are distributed evenly. For instance, in the case of a parabolic mirror, the light rays are reflected towards the focus, which is located on the axis of symmetry; finding this axis allows for the accurate placement of sensors or equipment at the focus point to maximize performance.