Question

The combined equation of the two lines passing through the origin angle 45° & 135° with the positive x axis is​

Answer 1

Answer:

Solution:

The auxiliary equation of the lines given by ax2 + 2hxy + by2 = 0 is bm2 + 2hm + a = 0. Since one of the lines bisects an angle between the coordinate axes, that line makes an angle of 45° or 135° with the positive direction of X-axis.

Explanation:

Line passes through the point (1,5).

It makes angle 135°with x-axis. Therefore,

Slope, m = tan135° =−1

Equation of line is

y−y 1 =m(x−x 1)y−5=−1(x−1)y−5=−x+1x+y−6=0

Hence, x+y-6=0 is correct.

Answer 2

The combined equation of the two lines is x² - y² = 0.

Given:

The combined equation of the two lines passing through the origin angle 45° & 135° with the positive x-axis is​  

Solution:

Formula used:

The slope of a line (m) = tan θ

Where θ is the angle that the line makes with the positive x-axis

From the data,

The line makes 45° with the positive x-axis  

The slope of the line (m₁) = tan 45° = 1  

And the line passes through the origin (0, 0)

Let the equation of the line is y = m₁x + c

=> y = (1) (x) + c    [ ∵ m₁ = 1 ]

=> y = x + c    

=> 0 = c               [ ∵ the line passes through (0, 0) ]

Hence, equation line (1) is y = x

=> x - y = 0 ----- (1)

It is also given that the line makes 135° with the x-axis  

The slope of the line (m₂) = tan 135° = - 1

Let the equation of the line is y = m₂x + c

=> y = (-1) (x) + c    [ ∵ m₂ = 1 ]

=> y = - x + c    

=> 0 = c               [ ∵ the line passes through (0, 0) ]

Hence, equation line (1) is y = - x

=> x + y = 0 ----- (2)  

The combined equation of two lines is calculated as follows

=> (x - y) (x + y) = 0

=> (x² - y²) = 0

     

Therefore,

The combined equation of the two lines is x² - y² = 0.

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