Name: Applications of Section Formula
Uploaded: 2019-07-19
Duration: 14 min 56 s
Description: Thorough with the section formula in three-dimensional geometry? This video will show you how the section formula can be applied in various situations in geo...

Question

The point $P$ is the reflection of the point $(- 7, 5)$ in the line $5 x - 3 y = 18$ . Find the coordinates of $P$ . You must show all your working.

Accepted Answer

Let $(- 7, 5)$ be $A$

Given line $5 x - 3 y = 18$ ...(i)

$\Rightarrow 3 y = 5 x - 18$

$\Rightarrow y = \frac{5}{3} x - 6$ , so the gradient is $\frac{5}{3}$

A line perpendicular to $5 x - 3 y = 18$ has a gradient $- \frac{3}{5}$ , since for perpendicular lines $m_{1} \times m_{2} = - 1$

The equation of the line which is perpendicular to $5 x - 3 y = 18$ and passes through $(- 7, 5)$ with slope $m = - \frac{3}{5}$ is $y - 5 = - \frac{3}{5} (x - (- 7))$

$\Rightarrow 5 y - 25 = - 3 x - 21$

$\Rightarrow 3 x + 5 y = 4$ ...(ii)

Multiplying (i) by $5$ and (ii) by $3$ then adding, we get

$34 x = 102$

$\Rightarrow x = 3$

From (ii), $3 (3) + 5 y = 4$

$\Rightarrow y = - 1$

The two lines intersect at $(3, - 1)$ (let $C$ )

Since $5 x - 3 y = 18$ is the perpendicular bisector of the line joining $(- 7, 5)$ to $P$ , using vectors

$\vec{A C} = (\begin{matrix} 3 - - 7 \\ - 1 - 5 \end{matrix}) = (\begin{matrix} 10 \\ - 6 \end{matrix})$

$\Rightarrow \vec{A C} = \vec{C P}$

$\Rightarrow \vec{C P} = (\begin{array}{l} 10 \\ - 6 \end{array})$

$∴$ The coordinates of $P$ are $(13, - 7)$ .

The point $P$ is the reflection of the point $(- 7, 5)$ in the line $5 x - 3 y = 18$ . Find the coordinates of $P$ . You must show all your working.

Important Questions on Coordinate Geometry

Learn and Prepare for any exam you want !

The point P is the reflection of the point (-7,5) in the line 5x-3y=18. Find the coordinates of P. You must show all your working.

Important Questions on Coordinate Geometry

Learn and Prepare for any exam you want !

The point $P$ is the reflection of the point $(- 7, 5)$ in the line $5 x - 3 y = 18$ . Find the coordinates of $P$ . You must show all your working.