EXPLORING OF THE RELATIONSHIP BETWEEN A CIRCLE AND A HYPERBOLA

Abstract

When two straight lines pass through the center of a circle with diameter $A A^{\prime}$ and are orthogonal to each other, and two points $Q$ and $Q^{\prime}$ on the circle's circumference are symmetrical with respect to the straight line passing through $A$ and $A^{\prime}$, the intersection point of the two lines $A Q$ and $A^{\prime} Q^{\prime}$ will lie on the tangent of the hyperbola at $A$ or $A^{\prime}$ on the circle. This theorem holds as a result of the fact that the product of the slopes of $A Q$ and $A^{\prime} Q^{\prime}$ is equal to one. This geometric concept can be effectively illustrated through "proofs without words (PWW)" using the ratio of similitude of right triangles.

Received: August 15, 2023
Accepted: October 3, 2023