Question

The equation of line passing through the point \((-5,4)\) and making the intercept of length \((2 / \sqrt{5})\) between the lines \(x+2 y-1=0\) and \(x+2 y+1=0\) is \(\ldots \ldots\) (a) \(2 \mathrm{x}-\mathrm{y}+4=0\) (b) \(2 x-y-14=0\) (c) \(2 \mathrm{x}-\mathrm{y}+14=0\) (d) None of these

Short answer

The equation of the line passing through the point \((-5,4)\) and making the intercept of length \((2 / \sqrt{5})\) between the lines \(x+2 y-1=0\) and \(x+2 y+1=0\) is (d) None of these.

Step-by-step solution

Find the slope of the given lines

: We are given two lines: Line 1: \(x + 2y - 1 = 0\) Line 2: \(x + 2y + 1 = 0\) The slope-intercept form of a line is: \(y = mx + b\), where m is the slope and b is the y-intercept. Rewriting the given lines in slope-intercept form, we get: Line 1: \(y = (-1/2)x + (1/2)\) Line 2: \(y = (-1/2)x - (1/2)\) The slope of both given lines is \(-1/2\). Since both lines have the same slope, they are parallel to each other.

Determine the slope of the required line

: The required line should have the same slope as the given lines since intercept of length \((2/\sqrt{5})\) is between the parallel lines. So, Required line slope: \(m = -1/2\)

Write the equation of the required line in point-slope form

: We have the slope of the required line, and we are given a point it passes through, which is \((-5, 4)\). The point-slope form of a line is: \(y - y_1 = m(x - x_1)\) Plugging in the given values and the calculated slope: \(y - 4 = -\frac{1}{2}(x - (-5))\) Now, let's simplify the equation: \(y - 4 = -\frac{1}{2}(x + 5)\)

Choose the correct answer from the given options

: Now, convert the simplified equation into the standard form and compare it with the given options. \(y - 4 = -\frac{1}{2}x -\frac{5}{2}\) \(2y - 8 = -x -5\) \(2x + 2y - 3= 0\) Comparing this with the given options, we see that the correct answer is not among the given options, so the answer is: (d) None of these

Key concepts

Slope-Intercept Form

The slope-intercept form is a very common way to express linear equations. It's formulated as \( y = mx + b \), where \( m \) represents the slope of the line and \( b \) denotes the y-intercept. The slope, \( m \), tells us how steep the line is. It's the rate at which the line ascends or descends as we move from left to right on the graph.
This form is particularly handy because:

• It instantly gives us the slope of the line.
• We can easily identify the y-intercept, where the line crosses the y-axis.
• Simplifies graphing the line.

In the provided problem, rearranging the equations like \( x + 2y - 1 = 0 \) into \( y = -\frac{1}{2}x + \frac{1}{2} \) helped identify that the slope of the lines is \(-\frac{1}{2}\). This consistency of the slope hinted that these lines are indeed parallel. This insight was crucial in establishing the line's needed equation passing through a given point.

Parallel Lines

Parallel lines are intriguing in geometry. They never meet and maintain a constant distance from each other. What's unique about them? They share the same slope. This is what makes them parallel!
In the exercise, both given lines had an identical slope of \(-1/2\). Recognizing this is key when you are working with parallel lines because if you need to find a line that is parallel to a given line, you must ensure it has the same slope.
This principle was applied in our solution, aiding the determination of the required line's slope. Even though the two lines had different y-intercepts, their slopes bind them into parallelism, providing a foundational insight into creating the equation for the line passing a given point in the problem.

Point-Slope Form

Sometimes, you know a point through which a line passes and its slope. In such cases, the point-slope form is invaluable. It's expressed as \( y - y_1 = m(x - x_1) \), where \((x_1, y_1)\) is a point on the line and \( m \) denotes the slope.
This form is particularly useful when:

• You have a specific point through which the line passes.
• The slope of the line is known or calculated.
• You want to derive or rearrange into other forms of linear equations.

In our problem, using the point \((-5, 4)\) and the determined slope \(-1/2\), this form was utilized to find the line's equation. By substituting \( x_1 =-5 \), \( y_1 =4 \), and \( m = -1/2 \) into the formula:
\( y - 4 = -\frac{1}{2}(x + 5) \).
This form helped effectively transition through the problem, leading to the required line equation even though none of the answer options matched exactly, displaying the practical application of this form.