Analytical Geometry> find the angle between the two st lines 3...

find the angle between the two st lines 3x=4y+7 and 5y=12x+6 and also the equations to the two lines which pass through the point (4,5) and make equal angles with the two given lines

Tejas Jaiswal , 10 Years ago

Grade 12th pass

1 Answers

Jitender Singh

Last Activity: 10 Years ago

Ans:
Hello student,
Please find answer to your question below

$4y = 3x - 7$ …...........(1)
$5y = 12x + 6$ ............(2)
$m_{1} = \frac{3}{4}, m_{2} = \frac{12}{5}$
Angle b/w two lines:
$tan\theta = |\frac{m_{1}-m_{2}}{1+m_{1}m_{2}}|$
$tan\theta = |\frac{\frac{3}{4}-\frac{12}{5}}{1+\frac{3}{4}.\frac{12}{5}}|$
$tan\theta = |\frac{-33}{56}| = \frac{33}{56}$
$\theta = tan^{-1}\frac{33}{56}$
Let the slope of the lines be ‘m’ which make equal angle with these lines
Then
$\frac{m-\frac{3}{4}}{1+m.\frac{3}{4}} = \pm \frac{m - \frac{12}{5}}{1+m.\frac{12}{5}}$
$(4m-3)(12m+5) = \pm (3m+4)(5m-12)$
$(48m^{2}-16m-15) = \pm (15m^{2}-16m-48)$
+ve sign is not possible
$63m^{2}-32m-63 = 0$
$\Rightarrow m =\frac{9}{7}, \frac{-7}{9}$
Equation of line
$(y-5) = \frac{-7}{9}(x-4)$
$(y-5) = \frac{9}{7}(x-4)$