Introduction to Chord of Contact and Tangents to a Circle

In this article, we will delve into the fascinating world of circle geometry, specifically focusing on the concept of chord of contact and tangents to a circle. We will collaborate with the mathematical principles to solve a complex problem and provide insights for high school students and anyone interested in the geometric properties of circles.

Definition and Key Formulas

The chord of contact is a line that touches a circle at the points where tangents are drawn from an external point to the circle. The formula to find the equation of the chord of contact for a circle with the equation (x^2 y^2 r^2) and a point ((x_1, y_1)) outside the circle is given by:

[xx_1 yy_1 r^2]

Solving the Problem

Consider the circle (x^2 y^2 21) and an external point ((-3, 4)). To find the equation of the chord of contact, we follow these steps:

Step 1: Identify Key Elements

The given circle equation is:

[x^2 y^2 21]

The point from which tangents are drawn is:

[(x_1, y_1) (-3, 4)]

The radius squared, (r^2), is 21.

Step 2: Apply the Chord of Contact Formula

Substitute the values into the chord of contact formula:

[xx_1 yy_1 r^2]

[x(-3) y(4) 21]

[-3x 4y 21]

Rearrange the equation to standard linear form:

[3x - 4y - 21 0]

Step 3: Verify and Explore Additional Geometric Properties

Let's explore the geometric properties related to this problem:

Additional Geometric Properties

Consider the circle (x^2 y^2 21). Suppose (A (-3, 4)) and tangents from this point touch the circle at points (B) and (C). The point (D) is the center of the circle at the origin (0,0).

The distance from the center to point (A) is (AC sqrt{(-3 - 0)^2 (4 - 0)^2} sqrt{9 16} 5). The radius of the circle is (sqrt{21}). The cosine of the angle (ACB) is (cos angle ACB frac{sqrt{21}}{5}). The chord (BD) is the shorter diagonal of the kite (ABCD) and intersects (AC) at (M). The distance from (C) to (M) is (frac{21}{5}). The equation of the longer diagonal (AC) is (y -frac{4}{3}x), or (4x 3y 0). Since (AC) is perpendicular to (BD), the equation of a line parallel to (BD) passing through (C) is (3x - 4y 0). The point (M) is at (frac{21}{5}) units from (C), so the equation of (BD) is (3x - 4y - frac{21}{5} sqrt{916} 0).

By solving these equations, we derive the final chord of contact equation:

[3x - 4y - 21 0]

Conclusion

The equation of the chord of contact for the circle (x^2 y^2 21) and the point ((-3, 4)) is (3x - 4y - 21 0). This problem showcases the application of circle geometry principles in solving practical mathematical scenarios.

Understanding these concepts not only enhances mathematical proficiency but also provides a deeper appreciation for the interconnectedness of geometric properties.