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Magazine Chapter 2: Problem 31
For what values of \(a\) is the distance between \(P(a, 3)\) and \(Q(5,2 a)\) greater than \(\sqrt{26} ?\)
Short Answer
Expert verified
The values of \(a\) are \(a < 0.4\) or \(a > 4\).
Step by step solution
01
Understanding the Distance Formula
The distance between two points \((x_1, y_1)\) and \((x_2, y_2)\) in a coordinate plane can be given by the formula: \[D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\].Given points are \( P(a, 3) \) and \( Q(5, 2a) \).
02
Apply the Distance Formula to Points P and Q
Substitute the coordinates of points \( P(a, 3) \) and \( Q(5, 2a) \) into the distance formula:\[D = \sqrt{(5 - a)^2 + (2a - 3)^2}\].
03
Expanding the Expression
Expand the expression inside the square root:\[(5 - a)^2 = 25 - 10a + a^2\]and \[(2a - 3)^2 = 4a^2 - 12a + 9\].Thus, \[D = \sqrt{a^2 - 10a + 25 + 4a^2 - 12a + 9}\].
04
Simplify the Expression
Combine like terms:\[D = \sqrt{5a^2 - 22a + 34}\].
05
Setting the Inequality for Distance
We need the distance to be greater than \(\sqrt{26} \), so set up the inequality:\[\sqrt{5a^2 - 22a + 34} > \sqrt{26}\].
06
Square Both Sides to Eliminate the Square Roots
By squaring both sides of the inequality, we get:\[5a^2 - 22a + 34 > 26\].
07
Rearrange and Simplify the Inequality
Rearrange the inequality:\[5a^2 - 22a + 34 - 26 > 0\]which simplifies to\[5a^2 - 22a + 8 > 0\].
08
Solve the Quadratic Inequality
To solve \(5a^2 - 22a + 8 > 0\), we find its roots using the quadratic formula \(a = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) where \(a = 5\), \(b = -22\), \(c = 8\):\[a = \frac{22 \pm \sqrt{(-22)^2 - 4 \cdot 5 \cdot 8}}{2 \cdot 5}\]\[a = \frac{22 \pm \sqrt{484 - 160}}{10}\]\[a = \frac{22 \pm \sqrt{324}}{10}\]\[a = \frac{22 \pm 18}{10}\].
09
Calculate the Roots
Calculate the roots \(a1\) and \(a2\):\[a1 = \frac{22 + 18}{10} = 4\]\[a2 = \frac{22 - 18}{10} = 0.4\].
10
Determine the Solution Interval
The quadratic \(5a^2 - 22a + 8\) is positive outside the interval defined by its roots \(a = 0.4\) and \(a = 4\), therefore, the values of \(a\) that satisfy the inequality are \(a < 0.4\) or \(a > 4\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Quadratic Inequality
A quadratic inequality involves an expression of the form \(ax^2 + bx + c > 0\), where the goal is to find the values of \(x\) that satisfy the inequality. In our exercise, we dealt with the inequality \(5a^2 - 22a + 8 > 0\). Solving this type of inequality involves several steps:
- First, we find the roots of the corresponding quadratic equation \(5a^2 - 22a + 8 = 0\) using the quadratic formula \(a = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).
- After finding the roots, which in this case were \(a = 0.4\) and \(a = 4\), these values divide the number line into intervals.
- We determine in which intervals the inequality is satisfied by testing a point from each interval. For this exercise, the quadratic inequality is positive for \(a < 0.4\) or \(a > 4\).
Coordinate Geometry
Coordinate geometry, also known as analytic geometry, involves representing geometrical shapes and their properties using a coordinate system. In this exercise, we worked with the points \((a, 3)\) and \((5, 2a)\). Understanding the following concepts is key:
- Coordinates are used to define the location of points on a plane.
- The distance between points is calculated using the distance formula, which relies on the differences between the x-coordinates and y-coordinates of the points.
- This form of geometry allows you to derive properties such as lengths, slopes, and areas in an algebraic manner.
Distance Between Points
The distance between two points in the coordinate plane is fundamental in coordinate geometry. The formula for finding this distance between points \((x_1, y_1)\) and \((x_2, y_2)\) is:\[D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]
In the given exercise, we used this formula to calculate the distance between points \((a, 3)\) and \((5, 2a)\). This involved:
In the given exercise, we used this formula to calculate the distance between points \((a, 3)\) and \((5, 2a)\). This involved:
- Substituting the coordinates into the distance formula:
- Expanding and simplifying the expression within the square root to derive the equation that was utilized for solving the quadratic inequality.
