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Chapter 6: Problem 64

Find the distance between the parallel lines. \(3 x-4 y=1\) \(3 x-4 y=10\)

Short Answer

Expert verified
The distance between the parallel lines \(3x - 4y = 1\) and \(3x - 4y = 10\) is \(\frac{9}{5}\).

Step by step solution

01

Gather Coefficients and Constants

The given equations are in the form ax + by = c. For both equations, the coefficients \(a = 3\), \(b = -4\). The constants \(c1\) and \(c2\) from the first and second equations are 1 and 10 respectively.
02

Implement the Distance Formula

Substitute these values into the distance formula, which is given as \(d = \frac{|c2-c1|}{\sqrt{a^2+b^2}}\). So, the distance \(d = \frac{|10 - 1|}{\sqrt{3^{2} + (-4)^{2}}}\).
03

Compute the Distance

Calculate the values in the equation. So, \(d = \frac{9}{\sqrt{9 + 16}} = \frac{9}{\sqrt{25}} = \frac{9}{5}\).

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Most popular questions from this chapter

Sketch the graph of the ellipse, using latera recta. \(\frac{x^{2}}{4}+\frac{y^{2}}{1}=1\)

Find the standard form of the equation of the ellipse with the given characteristics. Center: (2,-1)\(;\) vertex: \(\left(2, \frac{1}{2}\right) ;\) minor axis of length 2

Find the equation of an ellipse such that for any point on the ellipse, the sum of the distances from the point (2,2) and (10,2) is 36 .

Find the standard form of the equation of the ellipse with the given characteristics and center at the origin. Vertices: (卤7,0)\(;\) foci: (卤2,0)

An __________ is the set of all points \((x, y)\) in a plane, the sum of whose distances from two distinct fixed points, called _______ is constant.
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