Name: Family of Straight Lines: Example
Uploaded: 2019-07-19
Duration: 3 min 23 s
Description: Do you know how to find the equation of a straight line passing through the point of intersection of a family of two lines and also through a point? Let's find out.

Question

Find the equations of the two straight lines through

(1, 2)

forming two sides of a square of which

4 x + 7 y = 12

is one diagonal.

Accepted Answer

Let $A (1, 2)$ be the vertex of square $A B C D$ and $B D$ be one diagonal, whose equation is $4 x + 7 y = 12$

Here, we have to find the equations of sides $A B$ and $A D$ , each of which makes an angle of $45 °$ with line $4 x + 7 y = 12$

We know the equations of two lines passing through a point $(x_{1}, y_{1})$ and making an angle $α$ with the line whose slope is $m$ are

$y - y_{1} = \frac{m \pm \tan α}{1 \mp m \tan α} (x - x_{1})$

Here,

Equation of the given line is

$4 x + 7 y = 9$

$\Rightarrow y = - \frac{4}{7} x + \frac{9}{4}$

$∴ x_{1} = 1, y_{1} = 2, α = 45 °, m = - \frac{4}{7}$

So, the equations of the required sides are

$y - 2 = \frac{- \frac{4}{7} + \tan 45 °}{1 + \frac{4}{7} \tan 45 °} (x - 1)$ and $y - 2 = \frac{- \frac{4}{7} - \tan 45 °}{1 - \frac{4}{7} \tan 45 °} (x - 1)$

$\Rightarrow y - 2 = \frac{- \frac{4}{7} + 1}{1 + \frac{4}{7}} (x - 1)$ and $y - 2 = \frac{- \frac{4}{7} - 1}{1 - \frac{4}{7}} (x - 1)$

$\Rightarrow y - 2 = \frac{3}{11} (x - 1)$ and $y - 2 = \frac{- 11}{3} (x - 1)$

$\Rightarrow 3 x - 11 y + 19 = 0$ and $11 x + 3 y - 17 = 0$ .

Find the equations of the two straight lines through $(1, 2)$ forming two sides of a square of which $4 x + 7 y = 12$ is one diagonal.

Important Questions on Straight Lines

Learn and Prepare for any exam you want !

Find the equations of the two straight lines through (1,2) forming two sides of a square of which 4x+7y=12 is one diagonal.

Important Questions on Straight Lines

Learn and Prepare for any exam you want !

Find the equations of the two straight lines through $(1, 2)$ forming two sides of a square of which $4 x + 7 y = 12$ is one diagonal.