If three points $(h, 0),(a, b)$ and $(0, k)$ lie on a line, show that $\frac{a}{h}+\frac{b}{k}=1$.

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Geometry

NCERT Class 11 - Math

Chapter 10

Straight Lines

Section 1

Introduction

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Video Transcript

And this question we've been given three points. The first point is H comma zero. The second point is a comma B. And third point is zero comma. Okay, so now the question says that these three points lie or not like you simply have to prove that a upon edge plus B upon K. Is equal to what? Okay, so this is approving question over here. Now let us see that the point R is equal to h comma zero B is equal to a comma B. And C. B. The point that is equal to zero comma king. So these are the three point. Now it is already given that these three points are lying mono line. That means these three points are colonial, isn't it? Since they're lying on a straight line on the same line, that means they are colonial. Therefore I can say that the slope of a line A. B. Is equal to slope of the line Bc. Right? So now we know that the slope of the line is given by the formula and equal to buy two minus five and upon x two minus x one. So if you just use this formula were here and you find all the slope of the line A. B. Then what we get, I'm talking about slope of a B. Right slope of a B. That is M is given by M equal to b minus zero upon a minus h. Right? That is equal to be upon a minus H. And if we talk about the slope of the other line that the slope of B. C, then that is equal to I am equal to k minus beef upon zero minus eight. That means over here, M is equal to k minus B upon minus a. Now, as I say, since these points are cool in here, that means the slopes are equal to each other. So I can say be upon a minus age is equal to k minus b upon minus a. Now you can just cross multiply so we'll get minus a B is equal to a minus H multiplied by k minus video. You can just open up the brackets as well. So we're here we'll be getting minus a B. So called two K A minus a B minus hk plus H B. Okay, now you can just cancel the items or you can just combine the like terms and just cancel on minus a B from both sides of the equation. So now, what I'm getting over here is K A plus H B is equal to H K. Right now you can just divide put the sides by hk Okay, so divide by hk on both sides. So when you divide by hp on both the sides, what you will get a upon edge plus B upon K is equal to. So this was the same thing that we have to prove. So hence the giving a statement has been proved Hey?

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