find an equation of the tangent line to the circle x2 y2 4

Solution

Use implicit differentiation. We want: $\\frac{d}{dx}(x^2+y^2}=\\frac{d}{dx}(4)$ $\\frac{d}{dx}(x^2)+\\frac{d}{dx}{y^2} = 0$ $(2x^2)+2y \\frac{dy}{dx} = 0$ $2y \\frac{dy}{dx} = -2x$ $\\frac{dy}{dx} = \\frac{-x}{y}$ Now, substitute in the point $(\\sqrt{3}, 1)$, to get: $M = -\\sqrt{3}$. Now, we have the equation of the line as: $y=mx + b$. We find $b$ by substituting in values for $x$, $y$, and $m$: $1 = (-\\sqrt{3})(\\sqrt{3}) + b$ $b = 4$ Therefore, the equation of the line is: $y = -\\sqrt{3} x + 4$