Question

The ratio in which the x-axis divides the line segment joining the points $(6,4)$ and $(1, - 7)$ is $m:n,$ then $m + n =$

Answer 1

Hint : Here, we will frame the ratio as per the given data. Since, given that the ratio divides x-axis, the y-coordinate will be zero. Find the ratio and then as per the required answer add the numerator and the denominator of the ratio for the resultant answer.

Complete step-by-step answer :
Given that the x-axis divides the line – segment.
Therefore, let the coordinates be $(x,0)$ in the ratio in which the x-axis divides the line segment joining the points $(6,4)$ and $(1, - 7)$ is $k:1$
Using the bi-section formula-
$\dfrac{{ - 7k + 4}}{{k + 1}} = 0$
Now, do-cross multiplication and simplify-
$\Rightarrow - 7k + 4 = 0(k + 1)$
Anything multiplied with zero is equal to zero.
$\Rightarrow - 7k + 4 = 0$
Take the terms on the other side of the equation. Remember when you change the side of the equation, the sign of the term also changes. Positive terms become negative and vice-versa.
$\Rightarrow - 7k = - 4$
Negative signs from both the sides of the equation cancel each other.
$\Rightarrow 7k = 4$
Take the term on the other side and make the required value of “k” the subject. Remember when the term is in the multiplicative at one side changes its side, then it goes in the division on the opposite side.
$\Rightarrow k = \dfrac{4}{7}$
Now, the ratio “k” which is $m:n$
Here, $m = 4$
And $n = 7$
Add both the values,
$m + n = 4 + 7$
Simplify the above equation –
$m + n = 11$ is the required answer.
So, the correct answer is “ $m + n = 11$”.

Note : Be accurate while framing the first equation and giving the ratio as per the required data. Be careful while simplifying the equation. Be good in addition, subtraction, multiplication. Always remember zero multiplied with anything gives zero as the resultant value.
Ratio is the comparison between two numbers without any units.
Whereas, when two ratios are set equal to each other are called the proportion.
Four numbers a, b, c, and d are said to be in the proportion. If $a:b = c:d$ whereas, four numbers are said to be in continued proportion if the terms $$$$ $a:b = b:c = c:d$