Question

How do you find the equation of the line that passes through the origin and point \[\left( { - 3,6} \right)\] ?

Answer 1

Hint: Here in this question, we have to find the equation of the line that passes through the origin i.e., \[\left( {0,0} \right)\] and the given point. Find the equation by using the Point-Slope formula \[y - {y_1} = m\left( {x - {x_1}} \right)\] before finding the equation first we have to find the slope using the formula \[m = \dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}\]. On simplification to the point-slope formula we get the required solution.

Complete step by step solution:
Given the line which passes through the origin and the point.Hence the two points are \[\left( {0,0} \right)\] and \[\left( { - 3,6} \right)\]. Now, we have to find the equation of the line which passes through the points \[\left( {0,0} \right)\] and \[\left( { - 3,6} \right)\] by using the slope-point formula \[y - {y_1} = m\left( {x - {x_1}} \right)\]-------(1)
Before this, find the slope \[m\]in point-slope formula by using the formula \[m = \dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}\]
Where \[{x_1} = 0\], \[{x_2} = - 3\], \[{y_1} = 0\] and \[{y_2} = 6\] on substituting this in formula, then
 \[m = \dfrac{{6 - 0}}{{ - 3 - 0}}\]
\[ \Rightarrow \,\,\,m = \dfrac{6}{{ - 3}}\]
On simplification, we get
\[m = - 2\]
Now we get the gradient or slope of the line which passes through the points \[\left( {0,0} \right)\]. Substitute the slope m and the point \[\left( {{x_1},{y_1}} \right) = \left( {0,0} \right)\] in the point slope formula. Consider the equation (1)
\[y - {y_1} = m\left( {x - {x_1}} \right)\]
Where \[m = - 2\], \[{x_1} = 0\] and \[{y_1} = 0\] on substitution, we get
\[y - 0 = - 2\left( {x - 0} \right)\]
On simplification, we get
\[ \therefore\,\,y = - 2x\]

Hence, the equation of line that passes through the origin and the point \[\left( { - 3,6} \right)\] is \[y = - 2x\].

Note: To determine the equation of line we use the point-slope formula. While multiplying the terms we must take care of signs, and we should know about the sign conventions. On further simplification we must know about the simple arithmetic operations and the table of multiplication is needed.