Question
Prove that the product of the distance from any point on a hyperbola to its asymptotes is a constant.
Answer :
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To solve this problem, we will first express the equation of the hyperbola and its asymptotes, and then prove the constant product of distances from any point on the hyperbola to the asymptotes. ### 1. Equation of Hyperbola and Asymptotes Consider the standard form of the hyperbola: \[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \] The asymptotes of this hyperbola are the straight lines: \[ y = \pm \frac{b}{a} x \] These asymptotes pass through the origin and have slopes of \(\pm \frac{b}{a}\). ### 2. Distance from a Point to a Line The formula for the perpendicular distance from a point \((x_1, y_1)\) to a line \(Ax + By + C = 0\) is given by: \[ \text{Distance} = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}} \] For the asymptotes, the equations can be written in the form \(y - \frac{b}{a} x = 0\) and \(y + \frac{b}{a} x = 0\), which simplifies
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