Question

A square is formed by the two points of straight lines \(\mathrm{x}^{2}-8 \mathrm{x}+12=0\) and \(\mathrm{y}^{2}-14 \mathrm{y}+45=0 .\) A circle is inscribed in it. The centre of the circle is (a) \((6,5)\) (B) \((5,6)\) (c) \((7,4)\) (d) \((4,7)\)

Short answer

The center of the inscribed circle in the square formed by the two points of the straight lines \(x^2-8x+12=0\) and \(y^2-14y+45=0\) is (4, 7).

Step-by-step solution

Identify the coordinates of the intersecting points of the given straight lines

First, we will find the points where the two lines intersect. To do that, we will find the x-coordinates for line 1 and the y-coordinates for line 2, and then create an ordered pair with them. Line 1: \(x^2-8x+12=0\) Line 2: \(y^2-14y+45=0\) We will solve for the roots (x and y) of each line, using the quadratic formula: \(x = \frac{-b\pm\sqrt{b^2-4ac}}{2a}\)

Solve for the x-coordinates of the intersection points for Line 1

Using the quadratic formula for line 1: \(x = \frac{8\pm\sqrt{(-8)^2-4(1)(12)}}{2(1)}\) \(x = \frac{8\pm\sqrt{64-48}}{2}\) \(x = \frac{8\pm\sqrt{16}}{2}\) \(x = 2\) or \(x = 6\)

Solve for the y-coordinates of the intersection points for Line 2

Using the quadratic formula for line 2: \(y = \frac{14\pm\sqrt{(-14)^2-4(1)(45)}}{2(1)}\) \(y = \frac{14\pm\sqrt{196-180}}{2}\) \(y = \frac{14\pm\sqrt{16}}{2}\) \(y = 4\) or \(y = 10\) Now, we have the x-coordinates (2, 6) and the y-coordinates (4, 10) for the intersection points.

Find the four vertex coordinates of the square

Combine the x and y coordinates to get the 4 vertices: Vertex 1: (2, 4) Vertex 2: (2, 10) Vertex 3: (6, 4) Vertex 4: (6, 10)

Determine the center using the midpoint formula

Now we will use the midpoint formula to determine the center of the square. The midpoint formula is: Midpoint: \((\frac{x_1 + x_2}{2}, \frac{y_1+y_2}{2})\) Using the coordinates for opposite vertices, such as Vertex 1 (2, 4) and Vertex 4 (6, 10): Center: \((\frac{2 + 6}{2}, \frac{4 + 10}{2})\) Center: \((4, 7)\)

Match the center to the given options

Our calculated center of the inscribed circle is (4, 7). Looking at the given options, we can see that this matches option (d). So, the correct answer is (d) (4, 7).

Key concepts

Quadratic Equations

Quadratic equations are a fundamental aspect of algebra and appear frequently in various mathematical scenarios, including geometry. A quadratic equation is typically in the form of \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( a \) is not equal to zero.

The solutions to a quadratic equation, known as the roots, can be found using several methods, with the most common being factoring, completing the square, and the quadratic formula, which is \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). In the context of the inscribed circle in a square problem, the quadratic formula was used to determine the x and y coordinates of the square's vertices. It's crucial for students to understand that the roots can be real or complex and that they represent points where the parabola (graph of the quadratic equation) intersects the x-axis.

Exercise Improvement Advice

To enhance understanding, it is beneficial to demonstrate how the quadratic formula is derived from the standard form of a quadratic equation by completing the square. Additionally, visual aids showing the graph of the quadratic equation can help students better visualize the problem, especially in a geometric context such as finding the vertices of a square.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, involves the study of geometry using a coordinate system. This method combines algebra and geometry to describe the position of points, lines, and shapes in a two-dimensional plane. By using equations to represent geometric figures, we can perform algebraic calculations to solve geometric problems.

In our exercise, coordinate geometry allows us to work with the square and the inscribed circle using algebraic equations of lines and the concept of vertices (intersection points). Each vertex of the square is represented by an ordered pair \((x, y)\), with 'x' denoting the horizontal distance from the origin and 'y' indicating the vertical distance. This systematic approach is a powerful tool in solving a wide array of geometric problems and is foundational in advanced studies of geometry and related fields.

Exercise Improvement Advice

A great way to clarify coordinate geometry concepts is to use graph paper or digital graphing tools to plot the points, lines, and shapes. Annotating key elements, like the intersecting points that form the corners of the square, can further aid in visualizing and understanding the geometric configurations at play.

Midpoint Formula

The midpoint formula is a key concept in coordinate geometry, playing a vital role in determining the exact center point between two given points. It is particularly useful in problems involving symmetry, such as finding the center of a circle inscribed in a polygon. The formula is given by \( \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) \), where \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of the two points.

Applying the midpoint formula in our problem helps to locate the center of the square by using the coordinates of the diagonally opposite vertices. This center also coincides with the center of the inscribed circle, making the midpoint formula an indispensable tool for completing this exercise. Understanding the midpoint formula is crucial for students as it frequently appears in various aspects of geometry and problem-solving.

Exercise Improvement Advice

Students can improve their grasp of the midpoint formula by practicing with diverse sets of coordinates and by plotting these points and midpoints on a graph. Relating the midpoint to everyday contexts, like finding the halfway point between two locations, can also consolidate learning and make the concept more approachable and relevant.