Prove: Tangent to Hyperbola Bisected at Point of Tangency

[ankur162]

prove that the segment of the tangent to the hyperbola y=c/x . which is obtained b/w the co-ordinate axis is bisected at the point of tangency...

 

[daveb]

What have you worked out so far?

 

[tacman]

To prove that the segment of the tangent to the hyperbola y=c/x, which is obtained between the coordinate axes, is bisected at the point of tangency, we can use the properties of tangents to hyperbolas.

Firstly, we know that the tangent to a hyperbola at a point P on the curve is perpendicular to the radius of the hyperbola at that point. This means that the tangent forms a right angle with the radius at the point of tangency.

Now, let us consider the point of tangency of the tangent to the hyperbola y=c/x with the coordinate axes. Since the tangent is perpendicular to the radius at this point, the tangent and the coordinate axes form a right triangle.

Next, we can use the property of a right triangle that the perpendicular bisector of the hypotenuse passes through the midpoint of the hypotenuse. In this case, the hypotenuse is the segment of the tangent between the coordinate axes.

Therefore, the perpendicular bisector of the segment of the tangent between the coordinate axes passes through the point of tangency. This means that the segment is bisected at the point of tangency.

In conclusion, we have proved that the segment of the tangent to the hyperbola y=c/x, obtained between the coordinate axes, is bisected at the point of tangency. This is because the tangent is perpendicular to the radius at the point of tangency, and the perpendicular bisector of the segment of the tangent between the coordinate axes passes through the point of tangency.