Answer :
l₁ contains points of the form (kx, ky) and l₂ contains points of the form (-ky, kx), where k is a positive constant. This is true because of the given slopes and the fact that the lines intersect at the origin.
Lines l₁ and l₂ intersect at the origin, which means that the point (0,0) lies on both lines.
Given that l₁ has a slope of (y/x) where x > 0 and y > 0, we can conclude that any point on l₁ can be represented as (kx, ky) where k is a positive constant. This is because for any positive values of x and y, the ratio y/x will always yield a positive value. Therefore, l₁ contains all the points of the form (kx, ky).
Similarly, l₂ has a slope of -x/y. By substituting x = -y and y = x, we can represent any point on l₂ as (-ky, kx), where k is a positive constant. This is because when we substitute -y for x and x for y, the ratio -x/y will also yield a positive value. Hence, l₂ contains all the points of the form (-ky, kx).
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According to the given statement line l₁ must contain the point (x, y) where x > 0 and y > 0 and line l₂ must contain the point (-y, x) where x > 0 and y > 0 and they must be true..
To determine what must be true about lines l₁ and l₂, let's analyze the given information.
We are told that lines l₁ and l₂ intersect at the origin.
This means that the point (0, 0) lies on both lines.
The slope of line l₁ is given as (y/x), where x > 0 and y > 0.
This means that for every increase of x, there is a corresponding increase of y.
In other words, as we move along the line in the positive x-direction, the y-coordinate increases.
Therefore, line l₁ must contain the point (x, y) where x > 0 and y > 0.
The slope of line l₂ is given as -x/y.
This means that for every increase of y, there is a corresponding decrease of x.
In other words, as we move along the line in the positive y-direction, the x-coordinate decreases.
Therefore, line l₂ must contain the point (-y, x) where x > 0 and y > 0.
In conclusion, line l₁ contains the point (x, y) and line l₂ contains the point (-y, x).
The slopes and points on the lines are determined by the given conditions, ensuring that the lines intersect at the origin (0, 0).
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