Find the equation of a tangent to the unit circle at a given point $(a, b)$

To find the equation of a tangent to the unit circle at the point $(a, b)$, we start by noting that the unit circle is defined by:

$x^2 + y^2 = 1$

The slope of the radius at $(a, b)$ is $ \x0crac{b}{a} $, so the slope of the tangent line, being perpendicular to the radius, is:

$ -\x0crac{a}{b} $

Using the point-slope form of a line, the equation of the tangent line can be written as:

$ y – b = -\x0crac{a}{b}(x – a) $

Simplifying, we get:

$ bx + ay = 1 $

To find the equation of a tangent to the unit circle at the point $(a, b)$, note that the unit circle is given by:

$x^2 + y^2 = 1$

The radius to $(a, b)$ has a slope of $ x0crac{b}{a} $, so the tangent slope is:

$ -x0crac{a}{b} $

Using the point-slope form:

$ y – b = -x0crac{a}{b}(x – a) $

Rearranging gives:

$ bx + ay = 1 $

For the tangent line to the unit circle at $(a, b)$:

$ y – b = -x0crac{a}{b}(x – a) $

Which simplifies to:

$ bx + ay = 1 $

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