Expressing Coordinate b in Terms of a for a Point on a Perpendicular Line

This article will guide you through the process of finding an expression for the coordinate b in terms of a for a point R(a, b) given that the line perpendicular to QR is defined by the equation 5x - 2y 9. Follow along as we break down the steps to achieve this.

Step 1: Finding the Slope of the Given Line

We start by rewriting the given equation in the slope-intercept form y mx c, where m represents the slope.

Given equation: 5x - 2y 9.

Let's isolate y:

-2y -5x 9

y frac{5}{2}x - frac{9}{2}

Hence, the slope of the line is m frac{5}{2}.

Step 2: Determining the Slope of Line QR

Since the line perpendicular to QR has a slope of (frac{5}{2}), the slope of line QR, denoted as m_{QR}, will be the negative reciprocal of (frac{5}{2}).

m_{QR} -frac{2}{5}

This is because the product of the slopes of two perpendicular lines is (-1).

Step 3: Using the Slope Formula for Line QR

The slope of line QR can also be expressed using the coordinates of points Q(2, 3) and R(a, b) as follows:

m_{QR} frac{b - 3}{a - 2}

Step 4: Setting Up the Equation and Solving for b

Since we know the slope of line QR is (-frac{2}{5}), we can set up the equation:

(frac{b - 3}{a - 2} -frac{2}{5})

Cross-multiplying gives:

5(b - 3) -2(a - 2)

Expanding both sides:

5b - 15 -2a 4

Rearranging the equation to solve for b:

5b 2a 11

Finding b in terms of a by isolating it:

b frac{2a 11}{5}

This gives the expression for b in terms of a as the final result.

Alternative Method

Another way to derive the same result is by using the slope of a line perpendicular to the given line. The line perpendicular to 5x - 2y 9 can be written as 2x - 5y k. Given that this line passes through point (2, 3), we can find the value of k:

2(2) - 5(3) k

4 - 15 k

k -11

Hence, the equation of the line QR becomes:

2x - 5y -11

Since point (a, b) lies on this line, it must satisfy the equation:

2a - 5b -11

Rearranging to solve for b: