Perpendicular Lines Shortcut

You probably know how to solve this since solving perpendicular lines is very easy. Problem solving about this topic and its application was published few months ago in this page. But now let’s talk about speed and accuracy which is the prime purpose of a shortcut.

perpendicular line shortcut

Derivation:

Given a line with equation  ax+by+c=0. The equation of the line passing through point P(x_1,y_1)  and perpendicular to it can be solved as follows.

Two lines are perpendicular if and only if the product of their slopes is equal to -1.

Solving for the slope of the given line

ax+by+c=0

by=- ax-c

y=\displaystyle\frac{-ax-c}{b}

y=\displaystyle\frac{-ax}{b}-\displaystyle\frac{-c}{b}

The slope of the line in the form of y=mx+b is m. Hence, the slope of the line here is \displaystyle\frac{-a}{b}

Let m_2 be the slope of the perpendicular line.

Let m_1 be the slope of the given line.

Solving for the slope of the perpendicular line we have,

m_1m_2=-1

\displaystyle\frac{-a}{b}m_2=-1

m_2=\displaystyle\frac{b}{a}

Solving for the equation of perpendicular line we have

y-y_1=m_2(x-x_1)

y-y_1=\displaystyle\frac{b}{a}(x-x_1)

a(y-y_1)=bx-bx_1

ay-ay_1=bx-bx_1

bx-ay=bx_1-ay_1

This means that the equation of the line passing through (x_1,y_1) and perpendicular to the line ax_1+by_1+c=0  is  bx-ay=bx_1-ay_1.

Knowing this shortcut, you don’t need to calculate that much to solve for it. You just need to create your own mnemonics to effectively remember this shortcut so that you can pull it out of your head anytime you need it.

Worked Problem 1:

Find the equation of the line through (1,2) and perpendicular to the line 2x+y=5.

Solution:

Using the shortcut formula we have,

bx-ay=bx_1-ay_1

a=2,b=1,x_1=1,y_1=2

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

x-2y=-3

Or

x-2y+3=0

Worked Problem 2:

Find the equation of the line passing through  (-2,1)  and perpendicular to 2x-3y=4.

Solution:

Using the derived shortcut we have,

bx-ay=bx_1-ay_1

(-3)x-(2)y=(-3)(-2)-(2)(1)

-3x-2y=-8

3x+2y-8=0

Dan

Dan

Blogger and a Math enthusiast. Has no interest in Mathematics until MMC came. Aside from doing math, he also loves to travel and watch movies.
Dan

Latest posts by Dan (see all)

You may also like...

4 Responses

  1. Garrett says:

    Hello there! I know this is kinda off topic nevertheless I’d figured I’d ask. Would you be interested in exchanging links or maybe guest authoring a blog post or vice-versa? My website goes over a lot of the same subjects as yours and I believe we could greatly benefit from each other. If you’re interested feel free to shoot me an email. I look forward to hearing from you! Terrific blog by the way!

  2. Magnificent goods from you, man. I have understand your stuff previous to and you are just too great. I actually like what you’ve acquired here, really like what you’re stating and the way in which you say it. You make it enjoyable and you still care for to keep it sensible. I can not wait to read much more from you. This is really a tremendous web site.

  3. liverpool says:

    It`s really useful! Thank you!

  4. I would like to express my thanks to you for rescuing me from this scenario. Just after researching through the the net and obtaining opinions which are not beneficial, I assumed my entire life was well over. Existing devoid of the solutions to the issues you have sorted out by means of your main site is a critical case, as well as the kind which could have in a wrong way affected my career if I had not discovered your blog post. Your own personal ability and kindness in maneuvering every item was vital. I’m not sure what I would have done if I hadn’t come upon such a solution like this. It’s possible to at this time look ahead to my future. Thanks a lot so much for the expert and result oriented help. I will not be reluctant to propose your blog to any individual who wants and needs tips on this situation.

Leave a Reply

Your email address will not be published. Required fields are marked *