Find the equation of the tangent to the circle x^2+y^2-8 x-6 y+20=0 at the point (3,5)

Name: Problem 737, Chapter 42: Geometry
Uploaded: 2021-11-13T15:13:25-08:00
Duration: 50 s
Channel: Nick Johnson
Description: Find the equation of the tangent to the circle x^2+y^2-8 x-6 y+20=0 at the point (3,5)

step 1 Hello, so here we can go ahead and rewrite the equation of our given circle as x minus a negativ

Find the equation of the tangent to the circle x^2+y^2-8 x-6...

Key Concepts

Point-Slope Form

The point-slope form of a linear equation, expressed as y - y1 = m(x - x1), is useful for writing the equation of a line when a point on the line and its slope are known. In the context of finding a tangent, once the slope is determined using the perpendicularity property, the point-slope form provides a straightforward way to express the tangent line's equation.

Perpendicularity

The property of perpendicularity in geometry stipulates that a tangent line to a circle is perpendicular to the radius drawn to the point of tangency. This means that the slope of the tangent line is the negative reciprocal of the slope of the radius, which is a key relationship utilized when finding the equation of the tangent line.

Circle Equation

The circle equation represents a set of points in a plane that are equidistant from a fixed center. It is usually written in the standard form (x - a)^2 + (y - b)^2 = r^2, which clearly shows the center (a, b) and the radius r. Often, the equation of a circle given in general quadratic form must be manipulated by completing the square to reveal these characteristics.

Completing the Square

Completing the square is an algebraic technique used to convert a quadratic equation into a perfect square form. This method is particularly useful for rewriting the equation of a circle in standard form, allowing one to easily identify the center and radius of the circle.

Tangent Line

A tangent line to a circle is a line that touches the circle at exactly one point and is perpendicular to the radius at the point of tangency. This fundamental property of tangents is vital in determining the tangent's slope and subsequent equation.

Transcript

Please give Ace some feedback