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A walking path across a park is represented by the equation y=-4x+10. A new path will be built perpendicular to this path. The paths will intersect at the point (4,-6). Identify the equation that represents the new path. A. y=4x-22 B. y=1/4x-7 C. y=-4x+10 D. y=-1/4x-5

Sagot :

Given:

The equation of a line is

[tex]y=-4x+10[/tex]

A perpendicular line on the given line passes through the point (4,-6).

To find:

The equation of the perpendicular line.

Solution:

We have,

[tex]y=-4x+10[/tex]

On comparing this equation with slope intercept form [tex]y=mx+b[/tex], we get

[tex]m=-4[/tex]

So, the slope of the given line is -4.

We know that the product of slopes of two perpendicular lines is always -1.

[tex]m\times m_1=-1[/tex]

[tex](-4)\times m_1=-1[/tex]

[tex]m_1=\dfrac{-1}{-4}[/tex]

[tex]m_1=\dfrac{1}{4}[/tex]

The slope of the required line is [tex]\dfrac{1}{4}[/tex] and it passes through the point (4,-6). So, the equation of the line is

[tex]y-y_1=m_1(x-x_1)[/tex]

[tex]y-(-6)=\dfrac{1}{4}(x-4)[/tex]

[tex]y+6=\dfrac{1}{4}x-1[/tex]

[tex]y=\dfrac{1}{4}x-1-6[/tex]

[tex]y=\dfrac{1}{4}x-7[/tex]

Therefore, the correct option is B.

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