Question

find the angle between the lines . 3x+2y=6, 2x_3y =5​

Answer 1

$\begin{gathered}\begin{gathered}\bf \:Given \: 2 \: lines-\begin{cases} &\sf{3x + 2y = 6} \\ &\sf{2x - 3y = 5} \end{cases}\end{gathered}\end{gathered}$

$\begin{gathered}\begin{gathered}\bf \: To \: Find - \begin{cases} &\sf{angle \: between \: two \: lines}\end{cases}\end{gathered}\end{gathered}$

$\begin{gathered}\Large{\sf{{\underline{Formula \: Used - }}}}  \end{gathered}$

Let us consider a line ax + by + c = 0, then slope (m) of line is

$\boxed{\sf{\: m = - \dfrac{coefficient \: of \: x}{coefficient \: of \: y}}}$

Let us consider two lines having slope m and M respectively, and let p be the angle between the lines, then

$\boxed{\sf{\:tanp = \bigg |\dfrac{M - m}{1 + Mm} \bigg|}}$

$\large\underline{\sf{Solution-}}$

Given lines are

• 3x + 2y = 6 ------(1)

and

• 2x - 3y = 5 -----(2)

Now,

• Let m and M be slope of line (1) and line (2) respectively.

Consider, line (1)

• 3x + 2y = 6

$\rm :\longmapsto\:Slope \: of \: line, \: m \: = - \dfrac{3}{2}$

Consider, line (2)

• 2x - 3y = 5

$\rm :\longmapsto\:Slope \: of \: line, \: M = - \dfrac{2}{ - 3} = \dfrac{2}{3}$

• Let suppose that angle between line (1) and line (2) be p.

Therefore,

• We know,

$\rm :\longmapsto\:\:tanp = \bigg |\dfrac{M - m}{1 + Mm} \bigg|$

$\rm :\longmapsto\:\:tanp = \bigg |\dfrac{\dfrac{2}{3} + \dfrac{3}{2} }{1 + \dfrac{2}{3} \times \dfrac{ - 3}{2} } \bigg|$

$\rm :\longmapsto\:\:tanp = \bigg |\dfrac{\dfrac{4 + 9}{6} }{1 - 1} \bigg| = \bigg| \dfrac{13}{0} \bigg|$

$\rm :\longmapsto\:tanp = \infty$

$\bf\implies \:p \: = \: \dfrac{\pi}{2} \: or \: 90 \degree$

$\overbrace{ \underline { \boxed { \rm \therefore The \: angle\: between \: lines \: is \: \:p \: = \: \dfrac{\pi}{2} \: \: or \: \: 90 \degree)}}}$

Additional Information :-

• 1. If a line is parallel to x- axis, its slope is 0.

• 2. If a line is parallel to y- axis, its slope is not defined.

• 3. Two lines having slope m and M are parallel, iff m = M.

• 4. Two lines having slope m and M are perpendicular, iff M × m = - 1

• 5. Three points A, B and C are collinear if slope of AB = slope of BC = slope of CA.

Answer 2

Answer:

The answer will be right