Prove that the line through the point $\left(x_{1}, y_{1}\right)$ and parallel to the line $\mathrm{A} x+\mathrm{B} y+\mathrm{C}=0$ is
$$
\mathrm{A}\left(x-x_{1}\right)+\mathrm{B}\left(y-y_{1}\right)=0 .
$$

Answer

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Geometry

NCERT Class 11 - Math

Chapter 10

Straight Lines

Section 3

Various Forms of the Equation of a Line

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Video Transcript

That the line passing through the Point X1 comma Y one. And parallel to the line, A X plus B, Y plus c equals zero. Is of the form eight times x minus x one plus b times y minus vie one equal to zero. So now over here you have to find out the slope of this is a question that is X plus B by bless equal to zero. Okay, fine slope off X plus B Y plus equal to zero. Now, when you just find out the show, you just have to write it in the form of baikal two mx plus C. So that means B y is equal to minus x minus c. Or why will be equal to minus acorn B x minus C upon B. That means the above equation is of the form by going to mx Percy where um that is the slope of the line is equal to minus airborne. So now we know the slope of the given light. Now says both the lines are parallel to each other and we know that the slope of two panel lines are always equal to each other. Therefore, the slope of the required lying will also be equal to minus A upon be. Right now you have to find out the equation of the line. So now we know that the question of the line passing through the points X one comma by one and having a through of M is given by why minus Y one equal to M times x minus X one. Right. We just need to put in the values over here now. So I'm substituting the values we even get why minus Y one is supposed to slow pez minus airborne B Times X -X one. We can just multiply both the sides of the equation by B. So we'll get be kinds of I minus y one is a call to minus eight times x minus x one. Or you can just distribute the like you can have the brackets. So when you get beat by minus, sorry, B by minus B Y one zero to minus X plus e x one. Just rearrange the terms together. So we'll get a x minus a X one plus B, Y minus B. Y one is equal to zero. You can just take a common now, so a x minus X one plus B. Why minus Y one is equal to zero which is the required for, So hence it has been proved.

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