- The midpoint \(M(x_M, y_M)\) of segment \(PQ\) with \(P(x_1, y_1)\) and \(Q(x_2, y_2)\) is given by \(M\left(\frac{x_1+x_2}{2}, \frac{x_1^2+x_2^2-8(x_1+x_2)+30}{2}\right)\).
- The line of symmetry for a quadratic \(y=ax^2+bx+c\) is given by the formula \(x = -\frac{b}{2a}\).
- Substituting \(a=1\) and \(b=-8\) into the formula gives the equation of the line of symmetry: \(x = 4\).
- Final Answer: Part (a): \(M\left(\frac{x_1+x_2}{2}, \frac{x_1^2+x_2^2-8(x_1+x_2)+30}{2}\right)\); Part (b): \(x = 4\).
Explanation
- Analysis of the Quadratic Curve and Points P and Q
The quadratic curve is defined by the equation \(y = x^2 - 8x + 15\). Let the coordinates of points \(P\) and \(Q\) be \(P(x_1, y_1)\) and \(Q(x_2, y_2)\). Since both points lie on the parabola, they satisfy the equation \(y = x^2 - 8x + 15\), which means \(y_1 = x_1^2 - 8x_1 + 15\) and \(y_2 = x_2^2 - 8x_2 + 15\).
- Calculating the Coordinates of the Midpoint
The midpoint \(M(x_M, y_M)\) of the segment \(PQ\) has coordinates given by the averages of the x and y coordinates of \(P\) and \(Q\):
\[x_M = \frac{x_1 + x_2}{2}, \quad y_M = \frac{y_1 + y_2}{2}\]
Substituting the expressions for \(y_1\) and \(y_2\):
\[y_M = \frac{(x_1^2 - 8x_1 + 15) + (x_2^2 - 8x_2 + 15)}{2} = \frac{x_1^2 + x_2^2 - 8(x_1 + x_2) + 30}{2}\]
Using the relation \(x_1 + x_2 = 2x_M\), we substitute \(x_1 + x_2\) into the expression for \(y_M\):
\[y_M = \frac{x_1^2 + x_2^2}{2} - 4(2x_M) + 15 = \frac{x_1^2 + x_2^2}{2} - 8x_M + 15\]
The midpoint's y-coordinate is thus \(\left( \frac{x_1^2 + x_2^2}{2} - 8x_M + 15 \right)\), where \(x_M = \frac{x_1 + x_2}{2}\).
- Determining the Line of Symmetry
The line of symmetry of a parabola of the form \(y = ax^2 + bx + c\) is a vertical line passing through the vertex, given by \(x = -\frac{b}{2a}\). For the equation \(y = x^2 - 8x + 15\), we identify \(a = 1\) and \(b = -8\).
Calculating the line of symmetry:
\[x = -\frac{-8}{2(1)} = \frac{8}{2} = 4\]
The equation of the line of symmetry is \(x = 4\). This line is the vertical axis about which the parabola is symmetric, meaning any pair of points \(P\) and \(Q\) symmetric about this line will have a midpoint on this line, i.e., \(x_M = 4\).
Examples
The concept of a line of symmetry is crucial in architectural design and engineering, where balance and aesthetics are required. For example, when designing a bridge with a parabolic support, engineers must ensure the structure is symmetric about its central axis to distribute stress evenly. Calculating the axis of symmetry helps in placing support structures precisely, ensuring stability and safety for the construction project. If a support spans from point P to point Q on the arch, knowing that the midpoint lies on the axis of symmetry helps in balancing the load distribution.