# A point #P# moves between lines #y=0# and #y=mx# so that the area of quadrilateral formed by the two lines and perpendicular from #P# on these lines remains constant. Find the equation of locus of #P#?

The equation is of the locus is of type

Let us consider that equation of line

Point

Now it is evident that area of

and hence

=

=

and

Hence area of

and area of quadrilateral is

=

=

Hence equation of

This is the equation of a hyperbola.

Below is shown the graph for

graph{y^2-x^2+xy=10 [-20, 20, -10, 10]}

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The locus of point P can be found by understanding the geometric constraints involved. Let's denote the coordinates of point P as (x, y).

The area of the quadrilateral formed by the two lines y = 0 and y = mx and the perpendicular from P to these lines remains constant. Let A be this constant area.

The perpendicular from P to the line y = 0 will intersect the x-axis at a point with coordinates (x, 0), and the perpendicular from P to the line y = mx will intersect the line y = mx at a point with coordinates (x, mx). The distance between these two points will give the height of the quadrilateral, which is |mx|.

The length of the base of the quadrilateral (along the x-axis) is |x|.

Therefore, the area of the quadrilateral is given by the formula: Area = (1/2) * |mx| * |x| = (1/2) * |mx^2|.

Since this area is constant, we have |mx^2| = 2A.

This implies that |x^2| = 2A / |m|.

Hence, the locus of P is given by the equation |x^2| = 2A / |m|.

This represents a pair of curves, one in the first and third quadrants and the other in the second and fourth quadrants, symmetric about the y-axis.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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