Question

The points on \(\mathrm{Y}\) -axis at a distance 4 units from the line \(\mathrm{x}+4 \mathrm{y}=12\) are \(\ldots \ldots\) (a) \((3+\sqrt{14}, 0)(3-\sqrt{14}, 0)\) (b) \((-3-\sqrt{1} 7,0)(3+\sqrt{1} 7,0)\) (c) \((0,3+\sqrt{1} 7)\) (d) \((0,-3-\sqrt{17})(0,-3+\sqrt{17})\)

Short answer

The short answer for the points on the Y-axis at a distance 4 units from the line x+4y=12 is: (d) (0, -3-\(\sqrt{17}\))(0, -3+\(\sqrt{17}\))

Step-by-step solution

Write the equation in the given form

We have the equation x + 4y = 12.

Determine the slope and intercept of the given line

To do this, we rewrite the given equation in the slope-intercept form, y = mx + b, where m is the slope and b is the y-intercept. We have: x + 4y = 12 4y = -x + 12 y = -\(\frac{1}{4}\)x + 3 So, the slope (m) is -\(\frac{1}{4}\) and the y-intercept (b) is 3.

Find the two parallel lines 4 units away from the given line

Since these lines are parallel to the given line, they will have the same slope -\(\frac{1}{4}\). To find the y-intercept of these lines, we will use the following formula for the distance d between two parallel lines y = mx + b and y = mx + c, which is given by: d = \(\frac{|c - b|}{\sqrt{1 + m^2}}\) We are given d = 4 units, so we have the equation: 4 = \(\frac{|c - 3|}{\sqrt{1 + (-\frac{1}{4})^2}}\) 4 = \(\frac{|c - 3|}{\sqrt{\frac{17}{16}}}\) Now, multiply both sides by the square root term: 16 = |c - 3| * \(\sqrt{17}\)

Find the two possible values of the y-intercept for parallel lines

We have: 16 = |c - 3| * \(\sqrt{17}\) Divide both sides by \(\sqrt{17}\): 16 / \(\sqrt{17}\) = |c - 3| We have two possible cases: - (c - 3) = 16 / \(\sqrt{17}\) ⇒ c = 3 + 16 / \(\sqrt{17}\) - (c - 3) = -16 / \(\sqrt{17}\) ⇒ c = 3 - 16 / \(\sqrt{17}\)

Write the equations of the parallel lines

The equations of the parallel lines are: y = -\(\frac{1}{4}\)x + (3 + 16 / \(\sqrt{17}\)) or y = -\(\frac{1}{4}\)x + (3 - 16 / \(\sqrt{17}\))

Find the intersection points with the Y-axis

To find the intersection points with the Y-axis, we set x = 0 in the equations of the parallel lines. We get: y = -\(\frac{1}{4}\) * 0 + (3 + 16 / \(\sqrt{17}\)) ⇒ A = (0, 3 + 16 / \(\sqrt{17}\)) and y = -\(\frac{1}{4}\) * 0 + (3 - 16 / \(\sqrt{17}\)) ⇒ B = (0, 3 - 16 / \(\sqrt{17}\)) Hence, the points on the Y-axis at a distance of 4 units from the given line are: A = (0, 3 + 16 / \(\sqrt{17}\)) B = (0, 3 - 16 / \(\sqrt{17}\)) Comparing these values with the given options, we find that they match with the option: (d) (0, -3-\(\sqrt{17}\))(0, -3+\(\sqrt{17}\))

Key concepts

Coordinate Geometry

Coordinate geometry, often known as analytic geometry, is a system that uses numerical coordinates to define and analyze points, lines, and shapes on a plane. This method provides a bridge between algebra and geometry through graphs, allowing complex geometric problems to be solved using algebraic equations.

In this context, to determine the distance between parallel lines, we use a distinct formula derived from coordinate geometry principles. The process involves identifying the equations of the lines in a specific format, often the slope-intercept form, and applying knowledge about their slopes and y-intercepts to compute the distance. Furthermore, coordinate geometry encompasses the X and Y axes which are at the very heart of finding intersection points and establishing the position of lines and points in a plane.

Slope-Intercept Form

The slope-intercept form is particularly useful in coordinate geometry and is written as y = mx + b, where m represents the slope of the line, and b indicates the y-axis intersection point or y-intercept. This form provides a clear way to visualize how a line will look on a graph as it correlates directly with the line's ascent or descent and where it crosses the Y-axis.

When dealing with parallel lines, they will always have identical slopes, indicating they rise or fall at the same rate and will never converge. For the question at hand, converting the given equation to the slope-intercept form allows us to recognize the slope of the original line and subsequently identify parallel lines with the same slope, only shifted along the Y-axis.

Y-axis Intersection Points

The Y-axis intersection points, or y-intercepts, are essential elements in the analysis of lines on a coordinate plane. They represent the exact point where a line will cross the Y-axis, and this can be seen in the line's equation as the b value in the slope-intercept form. When x equals zero, the value of y will equal the y-intercept, thus revealing where the line meets the Y-axis.

In our exercise, the points of intersection with the Y-axis were found by setting x to zero in the equations of the parallel lines. This conclusive step showed us the precise points on the Y-axis that are a specific distance away from the given original line, leading to a solution that matches one of the provided options.