Welcome to Westonci.ca, the place where your questions are answered by a community of knowledgeable contributors. Experience the convenience of finding accurate answers to your questions from knowledgeable professionals on our platform. Get precise and detailed answers to your questions from a knowledgeable community of experts on our Q&A platform.

The line l is tangent to the circle with equation x² + y2 = 169 at the point P.
By thinking about the radius or otherwise, determine the gradient of the line l.


The Line L Is Tangent To The Circle With Equation X Y2 169 At The Point P By Thinking About The Radius Or Otherwise Determine The Gradient Of The Line L class=

Sagot :

Given:

The equation of a circle is

[tex]x^2+y^2=169[/tex]

A tangent line l to the circle touches the circle at point P(12,5).

To find:

The gradient of the line l.

Solution:

Slope formula: If a line passes through two points, then the slope of the line is

[tex]m=\dfrac{y_2-y_1}{x_2-x_1}[/tex]

Endpoints of the radius are O(0,0) and P(12,5). So, the slope of radius is

[tex]m_1=\dfrac{5-0}{12-0}[/tex]

[tex]m_1=\dfrac{5}{12}[/tex]

We know that, the radius of a circle is always perpendicular to the tangent at the point of tangency.

Product of slopes of two perpendicular lines is always -1.

Let the slope of tangent line l is m. Then, the product of slopes of line l and radius is -1.

[tex]m\times m_1=-1[/tex]

[tex]m\times \dfrac{5}{12}=-1[/tex]

[tex]m=-\dfrac{12}{5}[/tex]

Therefore, the gradient or slope of the tangent line l is [tex]-\dfrac{12}{5}[/tex] .