From where √2 comes in its answer? Why the answer is not y=3x-2?

In the figure below line p has the equation y =3x. Line q is below p, as shown , and q is parallel to p. Which of the following is an equation for q ?

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5 Answers

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  • 4 months ago

    Your diagram was shown not clearly. If 3 was the distance between p & q, then see the diagram below:

    tan(theta)=3/1

    Tri. ODC ~ Tri.OAB

    =>

    AB/OB=CD/OC

    =>

    AB/3=3/1

    =>

    AB=9

    =>

    OA=sqr(9^2+3^2)=3sqr(10)

    =>  

    q is passing through (0,-3sqr(10)), thus

    the equation of q is

    y+3sqr(10)=3x

    =>

    y=3x-3sqr(10).

    The problem is OB=3?

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  • Ray S
    Lv 7
    4 months ago

         The perpendicular to y=3x at the origin is y=-⅓x. And, at a distance of 3, this line intersects the circle x²+y²=9, of radius 3 centered on the origin, at point (9/√10 , -3/√10) ... which puts point (9/√10 , -3/√10) on line q at a distance of 3 from line p. Then, by the point-slope form of a line, we find that line q is actually y=3x-(30/√10) ... which is none of the options.

         So, either the problem is poorly written or you were asked to pick which of the given options would be the closest to being q. And, in that latter case, line q would need to have the form y=3x-b ... which points to options B and D ... But further, since the vertical distance from p to q would be greater than the perpendicular distance between p to q, b must be greater than 3 ... which eliminates option D ... Leaving only B as a real possibility since 3√2 > 3.

    ANSWER

    B. y=3x-3√2 ... is the closest to being q ... But, it's not q.

    https://www.wolframalpha.com/input/?i=distance+bet...

    —————————————————————————————

    Note: 

    The distance between y=3x and y=3x-3√2  is about 1.34

    https://www.wolframalpha.com/input/?i=distance+bet...

    —————————————————————————————

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  • Bryce
    Lv 7
    4 months ago

    If p= x and the distance between= 3, then q= x - 3√2.

  • Pope
    Lv 7
    4 months ago

    You must have left something out. Your attached image shows the answer choices, but not the actual question.

    First, if those axes have the same scale, then the line labeled p is nothing like y = 3x. They do not necessarily have the same scale though, so that might not be a problem here.

    Moving past that, the equations in options B, D, and E all represent lines that are parallel to line p. Were there any other conditions that need to be addressed? The sketch seems to indicate that 3 is the distance between lines p and q. If that is a condition, then I do not like any of the options.

    Let p be the line y = 3x. Below are equations for two lines. Each of them is parallel to line p, and separated from p by 3 units:

    y = 3x + 3√(10)

    y = 3x - 3√(10)

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  • 4 months ago

    We have the line (p) as y = 3x

    The line (q) has the same gradient as (p) and intercepts the y-axis below the origin,...i.e. a negative value

    Hence, our two options are y = 3x - 3 and y = 3x - 3√2

    We are told that the perpendicular distance between the lines is 3 units. The distance from the origin to the intercept is longer than 3 units

    Now, 3√2 > 3, hence y = 3x - 3√2 is line (q)

    You state, why the line is not y = 3x - 2. Well it's not one of the given options, also 2 < 3

    :)>

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