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

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

Options:

Correct Answer: x^2 - 2mxy + y^2 = 0

Solution:

The correct form of the equation representing the lines through the origin is x^2 - 2mxy + y^2 = 0.

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.

Pair of Lines – The equation represents two lines that intersect at the origin, characterized by their slopes m1 and m2.
Quadratic Form – The equation is a quadratic in terms of x and y, indicating a relationship between the two variables that forms two distinct lines.
Slope-Intercept Form – Understanding how the slopes m1 and m2 relate to the angles of the lines with respect to the x-axis.