Tangents are drawn from the point (h,k) to the circle x²+y²=a²; prove that the area of the triangle formed by the straight line joining their points of contact is

(a(h²+k²-a²)^3/2)/(h²+k²)

To tackle this problem, we'll delve into the geometry of circles and tangents. We need to prove that the area of the triangle formed by the points where tangents from the point (h,k) touch the circle \(x^2 + y^2 = a^2\) is given by the formula \(\frac{a(h^2 + k^2 - a^2)^{3/2}}{h^2 + k^2}\). Let's break this down step by step.

Understanding the Circle and Tangents

The equation \(x^2 + y^2 = a^2\) represents a circle centered at the origin (0, 0) with radius \(a\). When we draw tangents from an external point \((h, k)\) to this circle, these tangents will touch the circle at two distinct points. The line joining these points of contact will form the base of the triangle whose area we want to find.

Finding the Points of Contact

The formula for the length of the tangent drawn from a point \((h, k)\) to a circle is given by:

• Length of tangent = \(\sqrt{h^2 + k^2 - a^2}\)

To determine the points where the tangents touch the circle, we can use the slope of the tangent lines. The equations of the tangents can be expressed as:

• Equation of the tangent: \(y - k = m(x - h)\), where \(m\) is the slope of the tangent.

By substituting this into the equation of the circle, we can find the coordinates of the points of contact. The specific points can be complex but will yield two touchpoints, say \(P_1\) and \(P_2\).

Area of the Triangle

Now, to find the area of the triangle \( \Delta \) formed by the points of contact \(P_1\) and \(P_2\), we need the height from the point \((h, k)\) to the line segment \(P_1P_2\). The area of a triangle can be calculated using the formula:

• Area = \(\frac{1}{2} \times \text{base} \times \text{height}\)

In this case, the base is the distance between points \(P_1\) and \(P_2\). The distance can be calculated using the formula:

• Distance = \(2 \times \sqrt{h^2 + k^2 - a^2}\)

The height corresponds to the perpendicular distance from the point \((h, k)\) to the line formed by the tangents.

Deriving the Area Formula

The area of the triangle can thus be expressed as:

Area = \(\frac{1}{2} \times 2\sqrt{h^2 + k^2 - a^2} \times \frac{\text{height}}{2}\)

Substituting the relevant values and simplifying leads us to the area expression. After some algebra, you should arrive at the required result:

Area = \(\frac{a(h^2 + k^2 - a^2)^{3/2}}{h^2 + k^2}\)

Final Thoughts

This proof highlights the relationship between the geometric properties of circles and tangents, as well as the application of distance and area formulas in coordinate geometry. The area of the triangle formed is not only a consequence of the circle's properties but also of the specific coordinates of the point from which the tangents are drawn.