# Find the Joint Equation of the Pair of Lines Through the Origin Each of Which is Making an Angle of 30° with the Line 3x + 2y - 11 = 0 - Mathematics and Statistics

Find the joint equation of the pair of lines through the origin each of which is making an angle of 30° with the line 3x + 2y - 11 = 0

#### Solution

The slope of the line3x + 2y -11 = 0 is m1=3/2

Let m be the slope of one of the lines making an angle of 300 with the 3x + 2y -11= 0. The angle between the lines having slopes m and m1 is 30°

tan 30^@=|(m-m_1)/(1+mm_1)|,where tan 30^@=1/sqrt3

therefore 1/sqrt3=|(m-(-3/2))/(1+m(-3/2))|

On squaring both the sides, we get,

1/3=(2m+3)^2/(2-3m)^2

therefore (2-3m)^2=3(2m+3)^2

4-12m+9m^2=3(4m^2+12m+9)

9m^2-12m+4=12m^2+36m+27

3m^2+48m+23=0

This is the auxiliary equation of the two lines and their joint equation is obtained by putting m=y/x

∴ the joint equation of the two lines is

3(y/x)^2+48(y/x)+23=0

(3y)^2/x^2+(48y)/x+23=0

3

3y^2+48xy+23x^2=0

Concept: Pair of Straight Lines - Pair of Lines Passing Through Origin - Combined Equation
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