Question

What is the equation of the line that passes through the origin and is perpendicular to the line that passes through the following points: `(3,7),(5,8)`?

Answer 1

`y=-2x`

Explanation:

First of all, we need to find the gradient of the line passing through `(3,7)` and `(5,8)`

`"gradient"=(8-7)/(5-3)`

`"gradient"=1/2`

Now since the new line is PERPENDICULAR to the line passing through the 2 points, we can use this equation

`m_1m_2=-1` where the gradients of two different lines when multiplied should equal to `-1` if the lines are perpendicular to one another ie at right angles .

hence, your new line would have a gradient of `1/2m_2=-1`
`m_2=-2`

Now, we can use the point gradient formula to find your equation of the line
`y-0=-2(x-0)`
`y=-2x`

Answer 2

Equation of the passing through the origin and having slope = -2 is

`color(blue)(y = -2x " or " 2x + y = 0`

Explanation:

`A (3,7), B(5,8)`

`"Slope of line AB " = m = (y_b - y_a) / (x_b - x_a) = (8-7) / (5-3) = 1/2`

Slope of the perpendicular line = -1/m = -2#

Equation of the passing through the origin and having slope = -2 is

`(y - 0) = -2 (x - 0)`

`color(blue)(y = -2x " or " 2x + y = 0`

graph{-2x [-10, 10, -5, 5]}