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Chapter 6: Problem 63

Find the distance between the parallel lines. \(x+y=1\) \(x+y=5\)

Short Answer

Expert verified
The distance between the parallel lines is \(2\sqrt{2}\).

Step by step solution

01

Identify the coefficients A, B and constants C1, C2

In the equation of a line, \(Ax + By = C\), A and B are the coefficients of x and y respectively and C is a constant term. For the first line, \(x + y = 1\), A1 = B1 = 1 and \(C1 = 1\). For the second line, \(x + y = 5\), A2 = B2 = 1 and \(C2 = 5\). Since the coefficients of x and y are the same for the two lines, these lines are parallel.
02

Substitute the values into the distance formula

The formula for the distance between two parallel lines is \(d = \frac{|C2 - C1|}{\sqrt{A^2 + B^2}}\). Substituting the values from step 1 into this equation gives: \(d = \frac{|5 - 1|}{\sqrt{1^2 + 1^2}} = \frac{4}{\sqrt{2}}\).
03

Simplify the result

The expression \(\frac{4}{\sqrt{2}}\) can be simplified by rationalizing the denominator. Multiply both the numerator and denominator by \(\sqrt{2}\) to get \(d = 4\sqrt{2} / 2 = 2\sqrt{2}\).

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Most popular questions from this chapter

(d) Explain how the result of part (c) can be used as a sketching aid when graphing parabolas. Consider the parabola \(x^{2}=4 p y\) (a) Use a graphing utility to graph the parabola for \(p=1, p=2, p=3,\) and \(p=4\). Describe the effect on the graph when \(p\) increases. (b) Locate the focus for each parabola in part (a). (c) For each parabola in part (a), find the length of the latus rectum (see figure). How can the length of the latus rectum be determined directly from the standard form of the equation of the parabola?

Sketch the graph of the ellipse, using latera recta. \(5 x^{2}+3 y^{2}=15\)

Find the vertex, focus, and directrix of the parabola, and sketch its graph. \(y=\frac{1}{4}\left(x^{2}-2 x+5\right)\)

Identify the conic as a circle or an ellipse. Then find the center, radius, vertices, foci, and eccentricity of the conic (if applicable), and sketch its graph. \(6 x^{2}+2 y^{2}+18 x-10 y+2=0\)

A line segment through a focus of an ellipse with endpoints on the ellipse and perpendicular to the major axis is called a latus rectum of the ellipse. Therefore, an ellipse has two latera recta. Knowing the length of the latera recta is helpful in sketching an ellipse because it yields other points on the curve (see figure). Show that the length of each latus rectum is \(2 b^{2} / a\)
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