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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

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
Grade:12th pass

1 Answers

Jitender Singh IIT Delhi
askIITians Faculty 158 Points
6 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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