The equation of the pair of lines through the origin with slopes m1 and m2 is gi

The equation of the pair of lines through the origin with slopes m1 and m2 is given by:

The equation of the pair of lines through the origin with slopes m1 and m2 is given by:

Step 1: Understand that we want to find the equation of two lines that pass through the origin (0,0).
Step 2: Recognize that the slopes of the lines are given as m1 and m2.
Step 3: Recall that the general form of a line through the origin with slope m is y = mx.
Step 4: For two lines with slopes m1 and m2, we can write their equations as y = m1x and y = m2x.
Step 5: To combine these two equations into one equation, we can rearrange them into a standard form.
Step 6: The combined equation can be expressed as (y - m1x)(y - m2x) = 0.
Step 7: Expanding this gives us y^2 - (m1 + m2)xy + m1m2x^2 = 0.
Step 8: To match the form x^2 - 2mxy + y^2 = 0, we can set m = (m1 + m2)/2 and m1m2 = 1.
Step 9: Thus, the final equation representing the lines through the origin is x^2 - 2mxy + y^2 = 0.

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