Question

The line through point (m, -9) and (7, m) has slope m. The y – intercept of this line is
A.-18
B.-6
C.6
D.18

Answer 1

We will use the two – point slope form of the straight line given by $\left( {{y_2} - {y_1}} \right) = m\left( {{x_2} - {x_1}} \right)$ with slope m and passing through $\left( {{x_1},{y_1}} \right)$ and $\left( {{x_2},{y_2}} \right)$. Then, on putting the values of $\left( {{x_1},{y_1}} \right)$ and $\left( {{x_2},{y_2}} \right)$, we will get a quadratic equation in m and we will solve it for the value of m. Then, using the slope and point $\left( {{x_1},{y_1}} \right)$ through which the line passes, we will form the equation of line using slope – intercept form: $\left( {y - {y_1}} \right) = m\left( {x - {x_1}} \right)$. For y – intercept, we will put the value of x = 0 and hence, we will see which of the options matches the obtained answer.

Complete step-by-step answer:
We are given that a line passes through two points whose slope is m.
The points through which line passes are $\left( {{x_1},{y_1}} \right)$ and $\left( {{x_2},{y_2}} \right)$.
We are required to calculate the value of the y – intercept of the line.
We are given that the line passes through (m, -9) and (7, m).
Using two – point slope form: $\left( {{y_2} - {y_1}} \right) = m\left( {{x_2} - {x_1}} \right)$, we can calculate the value of m after putting the values of $\left( {{x_1},{y_1}} \right)$ and $\left( {{x_2},{y_2}} \right)$ given by
$\Rightarrow \left( {{y_2} - {y_1}} \right) = m\left( {{x_2} - {x_1}} \right) \\\ \Rightarrow \left( {m - \left( { - 9} \right)} \right) = m\left( {7 - m} \right) \\\ \Rightarrow m + 9 = 7m - {m^2} \\\ \Rightarrow {m^2} - 6m + 9 = 0 \\\$
Now, we get a quadratic equation in m. we can solve this equation for the value by factorization method as:
$\Rightarrow {m^2} - 6m + 9 = 0 \\\ \Rightarrow {m^2} - 3m - 3m + 9 = 0 \\\ \Rightarrow m\left( {m - 3} \right) - 3\left( {m - 3} \right) = 0 \\\ \Rightarrow \left( {m - 3} \right)\left( {m - 3} \right) = 0 \\\ \Rightarrow {\left( {m - 3} \right)^2} = 0 \\\ \Rightarrow m - 3 = 0 \\\ \Rightarrow m = 3 \\\$
Hence, the slope of the line is 3. Therefore, the points through which the line passes through are (3, -9) and (7, 3).
Now, we can calculate the equation of line using the slope as: $\left( {y - {y_1}} \right) = m\left( {x - {x_1}} \right)$
Therefore, the equation of the line is, after putting the value of m and $\left( {{x_1},{y_1}} \right)$, we get
$\Rightarrow y - 3 = 3\left( {x - 7} \right) \\\ \Rightarrow y - 3 = 3x - 21 \\\ \Rightarrow 3x - y = 18 \\\$
This is the equation of the line which passes through (3, -9) and (7, 3) with slope m = 3.
Now, we need to calculate the value of y – intercept of the line, hence, x – intercept of the line will be zero.
Therefore, putting x = 0, in the equation of the line $3x - y = 18$ , we get
$\Rightarrow 3\left( 0 \right) - y = 18 \\\ \Rightarrow - y = 18 \\\ \Rightarrow y = - 18 \\\$
Hence, - 18 is the y – intercept of the line and therefore, option (A) is correct.

Note: In such questions, we may get confused at many places since two different forms of straight line are used, i.e. two – point form and slope – intercept form of the straight lines. We can put (3, -9) as well in the slope – intercept form to calculate the equation of the line and even then, the equation would be same since $y - \left( { - 9} \right) = 3\left( {x - 3} \right) \Rightarrow y + 9 = 3x - 9 \Rightarrow 3x - y = 18$.