Is the Locus of Point P a Hyperbola When Area OMPN is Constant?

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The discussion focuses on proving that the locus of point P is a hyperbola when the area OMPN remains constant. It begins by establishing a coordinate system where one line, OM, is the y-axis and the other, ON, is defined by the equation y = mx. The perpendicular from P to OM is a horizontal line at y = y_0, while the perpendicular to ON is derived from the slope, leading to a second equation. By calculating the area of the quadrilateral formed by points O, M, P, and N as a function of the coordinates of P, and setting it equal to a constant, a relationship between x_0 and y_0 can be determined. This relationship ultimately demonstrates that the locus of point P is indeed a hyperbola.
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Homework Statement


From a point P, perpendiculars PM and PN are drawn to two fixed straight lines OM and ON. If the area OMPN, be constant, prove that the locus of P is a hyperbola.


How do we start?
 
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You can, without loss of generality, assume that one line, say OM, is the y-axis, and the other, ON, is given by y= mx for some number m. If P has coordinates (x_0, y_0) then the perpendicular to OM is the line y= y_0. The line through P perpendicular to ON is given by y= -(1/m)(x- x_0)+ y_0. You can find its length by finding x such that y= -(1/m)(x- x_0)+ y_0= mx, where the two lines cross. Find the area of that figure, as a function of x_0 and y_0, set it equal to a constant, and see what relation you get between x_0 and y_0.
 
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