# find the equations of the two straight lines which pass through the point (4,5)

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and make equal angles with the two straight lines 3x = 4y+7 and 12y = 5x+6 .

1)9x-7y=1;7x+9y=73.

2)2x-2y=1;x+9y=73.

3)9x-7y=1;7x+y=73.

4)9x-8y=1;7x+y=3.

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Solution:

The line passing through the point=(4,5)

The line equally inclined to the lines 3x=4y+7,5y=12x+6  means the angle between the required line and the two lines is equal assume the slop of the required equation is m1

The slope of the equation 3x=4y+7=>y=(3x)/4-7/4 so the slope of the equation is ¾

The slope of the equation 5y=12x+6=>y=(12x)/5+6/5 so the slope of the equation is 12/5

When slope of the two equations are formula for finding the angle between them is

a=tan-1(m1-m2)/1+m1m2=tan-1(m1-m2)/1+m1m2

=tan-1(m1-3/4)/1+3m1/4=tan-1(m1-12/5)1+12m1/5)

=>(4m1-3)/(4+3m1)=(5m1-12)/(5+12m1)

=>(20m1+48m12-15-36m1=20m1+15m12-48-36m1)

=>m12=1

=>m1=+1   or -1

so the two equations incline the given ttwo equations equally

those are passing throught he point  (4,5) and having the slope 1  and   -1

Those are y-x+1=0  and y+x-9=0