First slide

Equation of line in Straight lines

Question

The triangle formed by the lines $2 x + 3 y - 5 = 0, x - y + 4 = 0 and 6 x - 4 y + 1 = 0 is$

Moderate

A

Right angled
B

Equilateral
C

Isosceles
D

Right

Solution

Given lines are
$2 x + 3 y - 5 = 0 - - - - (1)$
$\begin{array}{l} x - y + 4 = 0 - - - (2) \\ 6 x - 4 y + 1 = 0 - - - - - - (3) \end{array}$
$The slope of the line a x + b y + c = 0 is - \frac{a}{b}$
$Observing the equations Options 1 and 3 their siopes are m_{1} = - \frac{2}{3}, m_{3} = \frac{3}{2}, whose product is -1$
Since the product of slopes of two sides is $- 1$ , the triangle formed by the given three lines is right angled triangle.
The slope of the second line is $m_{2} = 1$
If be the acute angle between (Options: 1) and (2)
$\begin{matrix} \tan θ = |\frac{m_{1} - m_{2}}{1 + m_{1} m_{2}}| \\ = \frac{- \frac{2}{3} - 1}{1 - \frac{2}{3}} \\ = - 5 \neq 1 (\tan 45 = 1) \end{matrix}$
Triangle is not isosceles; it is only right-angled triangle

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