Equation of a Straight Line 4 Units Away from the Origin with a Slope of -1

When dealing with equations of lines, sometimes you need to find specific lines that have certain properties, such as being a certain distance from the origin and having a particular slope. In this article, we will walk through the process of finding the equation of a straight line that is 4 units away from the origin and has a slope of -1. We will explore both the algebraic and geometric methods to arrive at the solution.

Algebraic Method

To find the equation of a straight line that is 4 units away from the origin with a slope of -1, we can use the point-slope form of the equation of a line, which is given by:

[]{y - y_1 m(x - x_1)}

where:

m is the slope, (x_1, y_1) is a point on the line.

Since the line is 4 units away from the origin (0, 0), we can use the distance formula to find the points that are 4 units away from the origin:

[]{sqrt{x^2 y^2} 4}

Squaring both sides gives us:

[]{x^2 y^2 16}

Given that the slope m -1, we can express y in terms of x:

[]{y -x b}

Substituting y -x b into the distance equation:

[]{x^2 (-x b)^2 16}

Expanding the equation:

[]{x^2 x^2 - 2bx b^2 16}

This simplifies to:

[]{2x^2 - 2bx b^2 - 16 0}

To find b for the line to be exactly 4 units away from the origin, this quadratic equation in x must have real solutions. The discriminant of this quadratic must be non-negative:

[]{D -2b^2 - 4 cdot 2 cdot (b^2 - 16) geq 0}

Calculating the discriminant:

[]{4b^2 - 8b^2 128 geq 0}

[]{-4b^2 128 geq 0}

[]{4b^2 leq 128}

[]{b^2 leq 32}

[]{b leq 4sqrt{2}}

Thus, the possible values for b are:

[]{b 4sqrt{2}} or b -4sqrt{2}

Final equations:

Substituting these values back into the line equation:

[]{y -x 4sqrt{2}} or y -x - 4sqrt{2}

These represent the lines parallel to each other at the specified distance from the origin.

Geometric Method

Alternatively, we can use a geometric approach to find the equations of the lines. Using the formula:

[]{C d cdot sqrt{m^2 1}}

Given d 4 and m -1, we can calculate:

[]{C 4 cdot sqrt{(-1)^2 1}} 4 cdot sqrt{2} pm 4sqrt{2}}

Substituting C into the line equation y mx C gives us:

[]{y -1x pm 4sqrt{2}} or y -1x 4sqrt{2} or y -1x - 4sqrt{2}}

This confirms the previous results.

Geometric Interpretation

The geometric interpretation involves considering the line as the tangent to a circle of radius 4 centered at the origin. The slope of the line perpendicular to the given line is the reciprocal of the slope of the given line, i.e., 1. This gives us specific points of tangency:

[]{x 4cos45° 2sqrt{2}, y 4sin45° 2sqrt{2}} or x 4cos225° -2sqrt{2}, y 4sin225° -2sqrt{2}}

The equations of the tangents at these points are given by:

[]{y - 2sqrt{2} -1(x - 2sqrt{2})} or y - (-2sqrt{2}) -1(x - (-2sqrt{2}))}

These simplify to:

[]{y - 2sqrt{2} -x 2sqrt{2}} or y 2sqrt{2} -x 2sqrt{2}}

Further simplification reveals the final equations:

[]{y -x 4sqrt{2}} or y -x - 4sqrt{2}}

Thus, the equations of the lines that are 4 units away from the origin with a slope of -1 are:

[]{y -x 4sqrt{2}} or y -x - 4sqrt{2}}