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3 years ago

Write the equation of the line passing through point (4, -1) and perpendicular to the line whose equation is 2x - y - 7 = 0.

1 answer:

7 0

Answer:

x + 2y + 4 =0

Explanation:

1) You know a point (4 - 1) and that the line is perpendicular to a given line.

2) From the equation of the perpendicular line whose equation was given, you determine the slope, for which you can use the slope-intercept form of the equation:

2x - y - 7 = 0 => y = 2x + 7.

The slope is the coeffiicient of x, which is 2.

Therefore, slope = 2.

3) The slope of the other line is the negative inverse of the slope of the perpedicular line:

slope = - 1/2

4) Now that you have the slope of your line and the point (4, - 1) you determine the equation with this procedure:

y - b
------- = slope
x - a

where a y b are the coordinates of the point (4, - 1) and the slope is -1/2

=>

y - (-1)
---------- = - 1/2
x - 4

5) Simplify

2(y + 1) = - (x - 4)

2y + 1 = - x + 4

2y + x +1 + 4 = 0

x + 2y + 4 =0 <---------- this is the equation searched.

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How do I find the equation of the perpendicular bisector between (-2,5) and (2,-1)?

Arturiano [62]

Answer:

$y = \frac{2}{3} x + 2$

Step-by-step explanation:

Start by finding the slope of the given line.

$\boxed{ slope = \frac{y _{1} - y_2 }{x_1 - x_2} }$

Slope of given line

$= \frac{5 - ( - 1)}{ - 2 - 2}$

$= \frac{5 + 1}{ - 4}$

$= \frac{6}{ - 4}$

$= - \frac{3}{2}$

A perpendicular bisector cuts through the line at its midpoint perpendicularly.

The product of the slopes of two perpendicular lines is -1.

Let the slope of the perpendicular bisector be m.

$- \frac{3}{2} m = - 1$

$m = - 1 \div ( - \frac{3}{2} )$

$m = - 1 \times ( - \frac{2}{3} )$

$m = \frac{2}{3}$

$y = \frac{2}{3}x + c$ , where c is the y-intercept.

To find the value of c, we need to substitute a pair of coordinates that lies on the perpendicular bisector into the equation. Since the perpendicular bisector passes through the midpoint of the given line, we can use the midpoint formula to find the coordinates.

$\boxed{midpoint = ( \frac{x _{1} + x _2}{2} , \frac{y_1 + y_2}{2} )}$

Midpoint of given line

$= ( \frac{ - 2+ 2}{2} , \frac{5 - 1}{2} )$

$=( \frac{0}{2} , \frac{4}{2} )$

= (0, 2)

$y = \frac{2}{3} x + c$

When x= 0, y= 2,

2= ⅔(0) +c

2= 0 +c

c= 2

Thus, the equation of the perpendicular bisector is $y = \frac{2}{3} x + 2$ .

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