Question

The line l is tangent to the circle with equation x² + y2 = 169 at the point P.

Answer 1

Given:

The equation of a circle is

$x^2+y^2=169$

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

$m=\dfrac{y_2-y_1}{x_2-x_1}$

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

$m_1=\dfrac{5-0}{12-0}$

$m_1=\dfrac{5}{12}$

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.

$m\times m_1=-1$

$m\times \dfrac{5}{12}=-1$

$m=-\dfrac{12}{5}$

Therefore, the gradient or slope of the tangent line l is $-\dfrac{12}{5}$ .