a point moves so that the product of it's directed distances from the lines 3x-4y+1=0 and 3x+4y-7=0 is 144. find the equation of it's locus, sketch the curve

Answer 1

The distance from a point P(m,n) to a line defined by Ax + By + C = 0 is |Am + Bn + C|/(A² + B²)^1/2. We want to find the equation for a point P(x,y) such that ±(3x-4y+1)(3x+4y-7)/25 = 144. Where the ± depends on how you are defining the "directed distance." This equation can be factored into a familiar form ± [(x-1)²/20² - (y-1)²/15²] = 1, which is the equation for a hyperbola centered at (1,1).