Find the equation of the tangent to the circle $x^2 + y^2 = 85$ at the point $\left(7, 6\right)$.
Equation of given circle: $\;$ $x^2 + y^2 = 85$
Comparing with the standard equation of circle $\;$ $x^2 + y^2 = r^2$ $\;$ gives
$r^2 = 85$
Given point: $\;$ $\left(x_1, y_1\right) = \left(7, 6\right)$
Equation of tangent to the circle $x^2 + y^2 = r^2$ at any given point $\left(x_1, y_1\right)$ is
$xx_1 + yy_1 = r^2$
$\therefore \;$ Required equation of tangent is
$7x + 6y = 85$