# The perpendicular from the origin to a line meets it at the point $( - 2,9).$ Find the equation of the line.

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Hint: Here first of all we will find the slope of the given points with reference to the origin and then the perpendicular slope with the help of the slope. Place the second slope in the line equation and simplify for the required answer.

Let the origin be $O(0,0)$
Then the slope of the line joining the origin $(0,0)$ and the point $( - 2,9)$ be ${m_1}$
Therefore, ${m_1} = \dfrac{{9 - 0}}{{ - 2 - 0}}$
Simplify the above expression –
$\Rightarrow {m_1} = \dfrac{9}{{ - 2}}$ .... (A)
Let the slope of other line be $= {m_2}$
By using the property the product of the slopes of the two perpendicular lines is always equal to $( - 1)$ .
Therefore, ${m_1}.{m_2} = ( - 1)$
Place value from the equation (A)
$\Rightarrow \left( {\dfrac{9}{{ - 2}}} \right).{m_2} = ( - 1)$
Do cross multiplication, where the numerator of one side is multiplied with the denominator of the other side.
$\Rightarrow {m_2} = \dfrac{{( - 1) \times ( - 2)}}{9}$
Product of two negative terms gives the positive term-
$\Rightarrow {m_2} = \dfrac{2}{9}$ .... (B)
Thus the equation of the line passing through the point $( - 2,9)$ and with the slope ${m_2}$ can be given by –
$(y - 9) = \dfrac{2}{9}(x + 2)$
Do cross-multiplication, where the numerator of one side is multiplied with the denominator on the other side of the equation.
$\Rightarrow 9(y - 9) = 2(x + 2)$
Multiply the constant inside the bracket and simplify the equation.
$\Rightarrow 9y - 81 = 2x + 4$
Bring all the terms on one side of the equation. When you move any term from one side to the other side of the equation, then sign of the terms also change. Positive term becomes negative and vice-versa.
$\Rightarrow 9y - 81 - 2x - 4 = 0$
Make pair of like terms and simplify –
$\Rightarrow 9y - 2x\underline { - 81 - 4} = 0 \\ \Rightarrow 9y - 2x - 85 = 0 \;$
This is the required solution.
So, the correct answer is “9y - 2x - 85 = 0”.

Note: Always remember that the slope of two perpendicular lines is always minus one. And be very careful while simplifying the equation and when you move any term from one side to another, the sign of the term also changes. Positive terms become negative and vice-versa.