What is the equation of the line that is perpendicular to X Y 7 and passes through (- 1 1 )?

We will learn how to find the equation of a line perpendicular to a line.

Prove that the equation of a line perpendicular to a given line ax + by + c = 0 is bx - ay + λ = 0, where λ is a constant.

Let m\(_{1}\) be the slope of the given line ax + by + c = 0 and m\(_{2}\) be the slope of a line perpendicular to the given line.

Then,

m\(_{1}\) = -\(\frac{a}{b}\) and m\(_{1}\)m\(_{2}\) = -1

⇒ m\(_{2}\) = -\(\frac{1}{m_{1}}\) = \(\frac{b}{a}\)

Let c\(_{2}\) be the y-intercept of the required line. Then its equation is

y = m\(_{2}\)x + c\(_{2}\)

⇒ y = \(\frac{b}{a}\) x + c\(_{2}\)

⇒ bx - ay + ac\(_{2}\) = 0

⇒ bx - ay + λ = 0, where  λ = ac\(_{2}\) = constant.

To get it more clear let us assume that ax + by + c = 0 (b ≠ 0) be the equation of the given straight line.

Now convert the ax + by + c = 0 in to slope-intercept form we get,

by = - ax - c     

⇒ y = - \(\frac{a}{b}\) x - \(\frac{c}{b}\)

Therefore, the slope of the straight line ax + by + c = 0 is (- \(\frac{a}{b}\)).

Let m be the slope of a line which is perpendicular to the line ax + by + c = 0. Then, we must have,

m × (- \(\frac{a}{b}\)) = - 1    

⇒ m = \(\frac{b}{a}\) 

Therefore, the equation of a line perpendicular to the line ax + by + c = 0 is

y = mx + c    

⇒ y = \(\frac{b}{a}\) x + c

⇒ ay =  bx +  ac

⇒ bx - ay+ k = 0, where k = ac, is an arbitrary constant.

Algorithm for directly writing the equation of a straight line perpendicular to a given straight line:

To write a straight line perpendicular to a given straight line we proceed as follows:

Step I: Interchange the coefficients of x and y in equation ax + by + c = 0.

Step II: Alter the sign between the terms in x and y of equation i.e., If the coefficient of x and y in the given equation are of the same signs make them of opposite signs and if the coefficient of x and y in the given equation are of the opposite signs make them of the same sign.

Step III: Replace the given constant of equation ax + by + c = 0 by an arbitrary constant.

For example, the equation of a line perpendicular to the line 7x + 2y + 5 = 0 is 2x - 7y + c = 0; again, the equation of a line, perpendicular to the line 9x - 3y = 1 is 3x + 9y + k = 0.

Note:

Assigning different values to k in bx - ay + k = 0 we shall get different straight lines each of which is perpendicular to the line ax + by + c = 0. Thus we can have a family of straight lines perpendicular to a given straight line.

Solved examples to find the equations of straight lines perpendicular to a given straight line

1. Find the equation of a straight line that passes through the point (-2, 3) and perpendicular to the straight line 2x + 4y + 7 = 0.

Solution:

The equation of a line perpendicular to 2x + 4y + 7 = 0 is

4x - 2y + k = 0 …………………… (i) Where k is an arbitrary constant.

According to the problem equation of the perpendicular line 4x - 2y + k = 0 passes through the point (-2, 3)

Then,

4 ∙ (-2) - 2 ∙ (3) + k = 0

⇒ -8 - 6 + k = 0

⇒ - 14 + k = 0

⇒ k = 14

Now putting the value of k = 14in (i) we get, 4x - 2y + 14 = 0

Therefore the required equation is 4x - 2y + 14 = 0.

2. Find the equation of the straight line which passes through the point of intersection of the straight lines x + y + 9 = 0 and 3x - 2y + 2 = 0 and is perpendicular to the line 4x + 5y + 1 = 0.

Solution:  

The given two equations are x + y + 9 = 0 …………………… (i) and 3x - 2y + 2 = 0 …………………… (ii)

Multiplying equation (i) by 2 and equation (ii) by 1 we get

                                                       2x + 2y + 18 = 0

                                                       3x  - 2y +   2 = 0

Adding the above two equations we get, 5x = - 20

⇒ x = - 4

Putting x = -4 in (i) we get, y = -5

Therefore, the co-ordinates of the point of intersection of the lines (i) and (ii) are (- 4, - 5).

Since the required straight line is perpendicular to the line 4x + 5y + 1 = 0, hence we assume the equation of the required line as

5x - 4y + λ = 0 …………………… (iii)

Where λ is an arbitrary constant.

By problem, the line (iii) passes through the point (- 4, - 5); hence we must have,

⇒ 5 ∙ (- 4) - 4 ∙ (- 5) + λ = 0  

⇒ -20 + 20 + λ = 0  

⇒ λ = 0.

Therefore, the equation of the required straight line is 5x - 4y = 0.

 The Straight Line

11 and 12 Grade Math

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What is the equation of the line that is perpendicular to X Y 7 and passes through (- 1 1 )?
What is the equation of the line that is perpendicular to X Y 7 and passes through (- 1 1 )?
What is the equation of the line that is perpendicular to X Y 7 and passes through (- 1 1 )?
What is the equation of the line that is perpendicular to X Y 7 and passes through (- 1 1 )?

What line is perpendicular to x + 3y = 6 and travels through point (1,5)?

Possible Answers:

Correct answer:

y = 3x + 2

Explanation:

Convert the equation to slope intercept form to get y = –1/3x + 2.  The old slope is –1/3 and the new slope is 3.  Perpendicular slopes must be opposite reciprocals of each other:  m1 * m2 = –1

With the new slope, use the slope intercept form and the point to calculate the intercept: y = mx + b or 5 = 3(1) + b, so b = 2

So y = 3x + 2

What line is perpendicular to

What is the equation of the line that is perpendicular to X Y 7 and passes through (- 1 1 )?
and passes through
What is the equation of the line that is perpendicular to X Y 7 and passes through (- 1 1 )?
?

Possible Answers:

What is the equation of the line that is perpendicular to X Y 7 and passes through (- 1 1 )?

What is the equation of the line that is perpendicular to X Y 7 and passes through (- 1 1 )?

What is the equation of the line that is perpendicular to X Y 7 and passes through (- 1 1 )?

What is the equation of the line that is perpendicular to X Y 7 and passes through (- 1 1 )?

What is the equation of the line that is perpendicular to X Y 7 and passes through (- 1 1 )?

Correct answer:

Explanation:

Convert the given equation to slope-intercept form.

What is the equation of the line that is perpendicular to X Y 7 and passes through (- 1 1 )?

What is the equation of the line that is perpendicular to X Y 7 and passes through (- 1 1 )?

What is the equation of the line that is perpendicular to X Y 7 and passes through (- 1 1 )?

The slope of this line is

What is the equation of the line that is perpendicular to X Y 7 and passes through (- 1 1 )?
. The slope of the line perpendicular to this one will have a slope equal to the negative reciprocal.

The perpendicular slope is 

What is the equation of the line that is perpendicular to X Y 7 and passes through (- 1 1 )?
.

Plug the new slope and the given point into the slope-intercept form to find the y-intercept.

What is the equation of the line that is perpendicular to X Y 7 and passes through (- 1 1 )?

What is the equation of the line that is perpendicular to X Y 7 and passes through (- 1 1 )?

What is the equation of the line that is perpendicular to X Y 7 and passes through (- 1 1 )?

So the equation of the perpendicular line is

What is the equation of the line that is perpendicular to X Y 7 and passes through (- 1 1 )?
.

What is the equation of a line that runs perpendicular to the line 2x + y = 5 and passes through the point (2,7)?

Possible Answers:

Correct answer:

–x/2 + y = 6

Explanation:

First, put the equation of the line given into slope-intercept form by solving for y. You get y = -2x +5, so the slope is –2. Perpendicular lines have opposite-reciprocal slopes, so the slope of the line we want to find is 1/2. Plugging in the point given into the equation y = 1/2x + b and solving for b, we get b = 6. Thus, the equation of the line is y = ½x + 6. Rearranged, it is –x/2 + y = 6.

Line m passes through the points (1, 4) and (5, 2). If line p is perpendicular to m, then which of the following could represent the equation for p?

Possible Answers:

Correct answer:

2x  y = 3

Explanation:

The slope of m is equal to   y2-y1/x2-x1  =  2-4/5-1 = -1/2                                  

Since line p is perpendicular to line m, this means that the products of the slopes of p and m must be 1:

(slope of p) * (-1/2) = -1

Slope of p = 2

So we must choose the equation that has a slope of 2. If we rewrite the equations in point-slope form (y = mx + b), we see that the equation 2x  y = 3 could be written as y = 2x – 3. This means that the slope of the line 2x – y =3 would be 2, so it could be the equation of line p. The answer is 2x – y = 3.

What is the equation for the line that is perpendicular to 

What is the equation of the line that is perpendicular to X Y 7 and passes through (- 1 1 )?
through point
What is the equation of the line that is perpendicular to X Y 7 and passes through (- 1 1 )?
?

Possible Answers:

What is the equation of the line that is perpendicular to X Y 7 and passes through (- 1 1 )?

What is the equation of the line that is perpendicular to X Y 7 and passes through (- 1 1 )?

What is the equation of the line that is perpendicular to X Y 7 and passes through (- 1 1 )?

What is the equation of the line that is perpendicular to X Y 7 and passes through (- 1 1 )?

What is the equation of the line that is perpendicular to X Y 7 and passes through (- 1 1 )?

Correct answer:

Explanation:

Perpendicular slopes are opposite reciprocals.

The given slope is found by converting the equation to the slope-intercept form.

What is the equation of the line that is perpendicular to X Y 7 and passes through (- 1 1 )?
 

The slope of the given line is

What is the equation of the line that is perpendicular to X Y 7 and passes through (- 1 1 )?
and the perpendicular slope is 
What is the equation of the line that is perpendicular to X Y 7 and passes through (- 1 1 )?
.

We can use the given point and the new slope to find the perpendicular equation. Plug in the slope and the given coordinates to solve for the y-intercept.

What is the equation of the line that is perpendicular to X Y 7 and passes through (- 1 1 )?

What is the equation of the line that is perpendicular to X Y 7 and passes through (- 1 1 )?

Using this y-intercept in slope-intercept form, we get out final equation:

What is the equation of the line that is perpendicular to X Y 7 and passes through (- 1 1 )?
.

Which line below is perpendicular to

What is the equation of the line that is perpendicular to X Y 7 and passes through (- 1 1 )?
?

Possible Answers:

What is the equation of the line that is perpendicular to X Y 7 and passes through (- 1 1 )?

What is the equation of the line that is perpendicular to X Y 7 and passes through (- 1 1 )?

What is the equation of the line that is perpendicular to X Y 7 and passes through (- 1 1 )?

What is the equation of the line that is perpendicular to X Y 7 and passes through (- 1 1 )?

What is the equation of the line that is perpendicular to X Y 7 and passes through (- 1 1 )?

Correct answer:

Explanation:

The definition of a perpendicular line is one that has a negative, reciprocal slope to another.

For this particular problem, we must first manipulate our initial equation into a more easily recognizable and useful form: slope-intercept form or

What is the equation of the line that is perpendicular to X Y 7 and passes through (- 1 1 )?
.

What is the equation of the line that is perpendicular to X Y 7 and passes through (- 1 1 )?

What is the equation of the line that is perpendicular to X Y 7 and passes through (- 1 1 )?

What is the equation of the line that is perpendicular to X Y 7 and passes through (- 1 1 )?

According to our 

What is the equation of the line that is perpendicular to X Y 7 and passes through (- 1 1 )?
formula, our slope for the original line is
What is the equation of the line that is perpendicular to X Y 7 and passes through (- 1 1 )?
. We are looking for an answer that has a perpendicular slope, or an opposite reciprocal. The opposite reciprocal of 
What is the equation of the line that is perpendicular to X Y 7 and passes through (- 1 1 )?
is
What is the equation of the line that is perpendicular to X Y 7 and passes through (- 1 1 )?
. Flip the original and multiply it by
What is the equation of the line that is perpendicular to X Y 7 and passes through (- 1 1 )?

Our answer will have a slope of

What is the equation of the line that is perpendicular to X Y 7 and passes through (- 1 1 )?
. Search the answer choices for 
What is the equation of the line that is perpendicular to X Y 7 and passes through (- 1 1 )?
in the 
What is the equation of the line that is perpendicular to X Y 7 and passes through (- 1 1 )?
position of the
What is the equation of the line that is perpendicular to X Y 7 and passes through (- 1 1 )?
equation.

What is the equation of the line that is perpendicular to X Y 7 and passes through (- 1 1 )?
is our answer. 

(As an aside, the negative reciprocal of 4 is

What is the equation of the line that is perpendicular to X Y 7 and passes through (- 1 1 )?
. Place the whole number over one and then flip/negate. This does not apply to the above problem, but should be understood to tackle certain permutations of this problem type where the original slope is an integer.)

If a line has an equation of 

What is the equation of the line that is perpendicular to X Y 7 and passes through (- 1 1 )?
, what is the slope of a line that is perpendicular to the line?

Possible Answers:

What is the equation of the line that is perpendicular to X Y 7 and passes through (- 1 1 )?

What is the equation of the line that is perpendicular to X Y 7 and passes through (- 1 1 )?

What is the equation of the line that is perpendicular to X Y 7 and passes through (- 1 1 )?

What is the equation of the line that is perpendicular to X Y 7 and passes through (- 1 1 )?

What is the equation of the line that is perpendicular to X Y 7 and passes through (- 1 1 )?

Correct answer:

Explanation:

Putting the first equation in slope-intercept form yields

What is the equation of the line that is perpendicular to X Y 7 and passes through (- 1 1 )?
.

A perpendicular line has a slope that is the negative inverse. In this case,

What is the equation of the line that is perpendicular to X Y 7 and passes through (- 1 1 )?
.

Which of the following is possibly a line perpendicular to 

What is the equation of the line that is perpendicular to X Y 7 and passes through (- 1 1 )?
?

Possible Answers:

What is the equation of the line that is perpendicular to X Y 7 and passes through (- 1 1 )?

What is the equation of the line that is perpendicular to X Y 7 and passes through (- 1 1 )?

What is the equation of the line that is perpendicular to X Y 7 and passes through (- 1 1 )?

What is the equation of the line that is perpendicular to X Y 7 and passes through (- 1 1 )?

What is the equation of the line that is perpendicular to X Y 7 and passes through (- 1 1 )?

Correct answer:

Explanation:

To start, begin by dividing everything by 

What is the equation of the line that is perpendicular to X Y 7 and passes through (- 1 1 )?
, this will get your equation into the format 
What is the equation of the line that is perpendicular to X Y 7 and passes through (- 1 1 )?
.  This gives you:

What is the equation of the line that is perpendicular to X Y 7 and passes through (- 1 1 )?

Now, recall that the slope of a perpendicular line is the opposite and reciprocal slope to its mutually perpendicular line.  Thus, if our slope is 

What is the equation of the line that is perpendicular to X Y 7 and passes through (- 1 1 )?
, then the perpendicular line's slope must be 
What is the equation of the line that is perpendicular to X Y 7 and passes through (- 1 1 )?
.  Thus, we need to look at our answers to determine which equation has a slope of 
What is the equation of the line that is perpendicular to X Y 7 and passes through (- 1 1 )?
.  Among the options given, the only one that matches this is 
What is the equation of the line that is perpendicular to X Y 7 and passes through (- 1 1 )?
.  If you solve this for 
What is the equation of the line that is perpendicular to X Y 7 and passes through (- 1 1 )?
, you will get:

What is the equation of the line that is perpendicular to X Y 7 and passes through (- 1 1 )?

Which of the following is the equation of a line perpendicular to the line given by:

What is the equation of the line that is perpendicular to X Y 7 and passes through (- 1 1 )?
?

Possible Answers:

What is the equation of the line that is perpendicular to X Y 7 and passes through (- 1 1 )?

What is the equation of the line that is perpendicular to X Y 7 and passes through (- 1 1 )?

What is the equation of the line that is perpendicular to X Y 7 and passes through (- 1 1 )?

What is the equation of the line that is perpendicular to X Y 7 and passes through (- 1 1 )?

What is the equation of the line that is perpendicular to X Y 7 and passes through (- 1 1 )?

Correct answer:

Explanation:

For two lines to be perpendicular their slopes must have a product of 

What is the equation of the line that is perpendicular to X Y 7 and passes through (- 1 1 )?
.
What is the equation of the line that is perpendicular to X Y 7 and passes through (- 1 1 )?
 and so we see the correct answer is given by 
What is the equation of the line that is perpendicular to X Y 7 and passes through (- 1 1 )?

What is the equation of a line perpendicular to the line defined by the equaiton:

What is the equation of the line that is perpendicular to X Y 7 and passes through (- 1 1 )?

Possible Answers:

What is the equation of the line that is perpendicular to X Y 7 and passes through (- 1 1 )?

What is the equation of the line that is perpendicular to X Y 7 and passes through (- 1 1 )?

What is the equation of the line that is perpendicular to X Y 7 and passes through (- 1 1 )?

What is the equation of the line that is perpendicular to X Y 7 and passes through (- 1 1 )?

What is the equation of the line that is perpendicular to X Y 7 and passes through (- 1 1 )?

Correct answer:

Explanation:

Perpendicular lines have slopes whose product is 

What is the equation of the line that is perpendicular to X Y 7 and passes through (- 1 1 )?
.

Looking at our equations we can see that it is in slope-intercept form where the m value represents the slope of the line,

What is the equation of the line that is perpendicular to X Y 7 and passes through (- 1 1 )?
.

In our case we see that

 therefore, 

What is the equation of the line that is perpendicular to X Y 7 and passes through (- 1 1 )?
.

Since 

What is the equation of the line that is perpendicular to X Y 7 and passes through (- 1 1 )?
 we see the only possible answer is 

What is the equation of the line that is perpendicular to X Y 7 and passes through (- 1 1 )?
.

What is the equation of the line that is perpendicular to X Y 7 and passes through (- 1 1 )?

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What is the equation of the line that is perpendicular to X Y 7 and passes through (- 1 1 )?

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What is the equation of the line that is perpendicular to X Y 7 and passes through (- 1 1 )?