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 [itex](x_0, y_0)[/itex] then the perpendicular to OM is the line [itex]y= y_0[/itex]. The line through P perpendicular to ON is given by [itex]y= -(1/m)(x- x_0)+ y_0[/itex]. You can find its length by finding x such that [itex]y= -(1/m)(x- x_0)+ y_0= mx[/itex], where the two lines cross. Find the area of that figure, as a function of [itex]x_0[/itex] and [itex]y_0[/itex], set it equal to a constant, and see what relation you get between [itex]x_0[/itex] and [itex]y_0[/itex].