Equation of Tangents to Circle at Given Points with Abscissa 1

In this article, we explore the equation of tangents to a circle defined by the equation x^2 y^2 - 10 0 at points where the x-coordinate is 1. This process involves a step-by-step breakdown of the equation transformation and substitution, followed by finding the equations of the tangents at specific points.

Standard Form of the Circle Equation

The given equation of the circle is x^2 y^2 - 10 0. To work with it more conveniently, we rewrite it in the standard form:

x^2 y^2 10

This indicates that the center of the circle is at the origin (0, 0), and the radius is sqrt{10}.

Finding Points on the Circle

We are interested in points where the x-coordinate (abscissa) is 1. Substituting x 1 into the equation of the circle, we get:

1^2 y^2 10

Simplifying, we find:

1 y^2 10

y^2 9

y 3 or y -3

Therefore, the points on the circle where the x-coordinate is 1 are (1, 3) and (1, -3).

Equations of Tangents

The general equation of a tangent to a circle x^2 y^2 r^2 at a point (x_0, y_0) is:

xx_0 yy_0 r^2

Given that r^2 10, we can find the tangents at the points (1, 3) and (1, -3).

Tangent at Point (1, 3)

Substituting x_0 1 and y_0 3 into the general equation:

x * 1 y * 3 10

x 3y 10

Tangent at Point (1, -3)

Substituting x_0 1 and y_0 -3 into the general equation:

x * 1 y * -3 10

x - 3y 10

Deriving Tangents Using Differentiation

Another method to derive these tangents is by differentiating the circle's equation to find the slope of the tangent at the given points. The equation of the circle is:

x^2 y^2 10

Using implicit differentiation:

2x 2y u2148 dy/dx 0

dy/dx -x/y

At point (1, 3), the slope dy/dx -1/3.

At point (1, -3), the slope dy/dx 1/3.

Using the point-slope form, the equation of the tangent at point (1, 3) is:

y - 3 -1/3(x - 1)

Simplifying:

3y - 9 -x 1

x 3y - 10 0

Similarly, for the tangent at point (1, -3) with slope 1/3:

y 3 1/3(x - 1)

Simplifying:

3y 9 x - 1

x - 3y - 10 0

Conclusion

The equations of the tangents to the circle x^2 y^2 - 10 0 at the points where the x-coordinate is 1 are:

x 3y 10 and x - 3y 10

This method can be generalized for any circle and any point, making it a powerful tool in geometry and calculus.