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Grade 12th passAnalytical Geometry

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

Question image for find the angle between the two st lines 3x=4y+7 an

Tejas Jaiswal

11 Years agoGrade 12th pass

1 Answer

Jitender Singh

11 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
$(y-5) = \frac{-7}{9}(x-4)$
$(y-5) = \frac{9}{7}(x-4)$

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