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

Answer

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Hint: We know that only two points are needed to draw a line but we have three points and all of them are lying on the same line.

To calculate slope(m) of a line formed by two points (p,q) and (r,s) we have the following formula:

$m=\dfrac{q-s}{p-r}$

We can use this formula to form an equation and show the desired result.

Complete step by step answer:

We know that if we calculate slope by using any two of the given three points they all must be equal.

Using $(h,0)$ and $(0,k)$ to calculate slope we have, $m=\dfrac{0-k}{h-0}\Rightarrow m=-\dfrac{k}{h}$

Using $(h,0)$ and $(a,b)$ to calculate slope we have, $m=\dfrac{0-b}{h-a}\Rightarrow m=\dfrac{-b}{h-a}$

Since, they all lie on the same line both of the above calculated slopes must be equal. Therefore, we have,

$-\dfrac{k}{h}=-\dfrac{b}{h-a}$

Cancelling -1 both sides we have,

$\dfrac{k}{h}=\dfrac{b}{h-a}$

After cross-multiplying we have,

$k(h-a)=h\cdot b$

Dividing both sides with $h\cdot k$ we have,

$\begin{align}

& \dfrac{1}{h}(h-a)=\dfrac{b}{k} \\

& \Rightarrow 1-\dfrac{a}{h}=\dfrac{b}{k} \\

\end{align}$

Now adding $\dfrac{a}{h}$ both sides we have,

$1=\dfrac{a}{h}+\dfrac{b}{k}$

Therefore, we can write $\dfrac{a}{h}+\dfrac{b}{k}=1$ .Hence, Proved.

Note: There can be alternative ways to solve this question. We can form the equation of the line with two of the given points. Now the equation of this line will satisfy the third point as it lies on this line. In this way we can also prove the given question.

We can use two points to calculate the slope of the line as mentioned earlier and the equation of a line is $y=mx+c$ where m=slope of the line and c is some constant. To find this ‘c’ we can plug in one the points in place of x and y. Therefore, we will have an equation of the line. Now again if we again plug in the remaining point and manipulate as earlier we will have our result.

To calculate slope(m) of a line formed by two points (p,q) and (r,s) we have the following formula:

$m=\dfrac{q-s}{p-r}$

We can use this formula to form an equation and show the desired result.

Complete step by step answer:

We know that if we calculate slope by using any two of the given three points they all must be equal.

Using $(h,0)$ and $(0,k)$ to calculate slope we have, $m=\dfrac{0-k}{h-0}\Rightarrow m=-\dfrac{k}{h}$

Using $(h,0)$ and $(a,b)$ to calculate slope we have, $m=\dfrac{0-b}{h-a}\Rightarrow m=\dfrac{-b}{h-a}$

Since, they all lie on the same line both of the above calculated slopes must be equal. Therefore, we have,

$-\dfrac{k}{h}=-\dfrac{b}{h-a}$

Cancelling -1 both sides we have,

$\dfrac{k}{h}=\dfrac{b}{h-a}$

After cross-multiplying we have,

$k(h-a)=h\cdot b$

Dividing both sides with $h\cdot k$ we have,

$\begin{align}

& \dfrac{1}{h}(h-a)=\dfrac{b}{k} \\

& \Rightarrow 1-\dfrac{a}{h}=\dfrac{b}{k} \\

\end{align}$

Now adding $\dfrac{a}{h}$ both sides we have,

$1=\dfrac{a}{h}+\dfrac{b}{k}$

Therefore, we can write $\dfrac{a}{h}+\dfrac{b}{k}=1$ .Hence, Proved.

Note: There can be alternative ways to solve this question. We can form the equation of the line with two of the given points. Now the equation of this line will satisfy the third point as it lies on this line. In this way we can also prove the given question.

We can use two points to calculate the slope of the line as mentioned earlier and the equation of a line is $y=mx+c$ where m=slope of the line and c is some constant. To find this ‘c’ we can plug in one the points in place of x and y. Therefore, we will have an equation of the line. Now again if we again plug in the remaining point and manipulate as earlier we will have our result.

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