logo

Equation of the Tangent to a Circle with Slope m 📂Geometry

Equation of the Tangent to a Circle with Slope m

Formula

The equation of the tangent line to the circle x2+y2=r2x^{2}+y^{2}=r^{2} with slope mm is as follows.

y=mx±rm2+1 y=mx \pm r\sqrt{m^{2}+1}

Proof

m.png

Let’s denote the equation of the line with slope mm as y=mx+ny=mx+n. Substituting into the equation of the circle and rearranging for x, we get

x2+(mx+n)2= r2x2+m2x2+2mnx+n2−r2= 0(1+m2)x2+2mnx+n2−r2= 0 \begin{align*} x^2+(mx+n)^2 =&\ r^2 \\ x^2+m^2x^2+2mnx+n^2-r^2 =&\ 0 \\ (1+m^2)x^2+2mnx+n^2-r^2 =&\ 0 \end{align*}

Since the circle and the line are tangent to each other, the discriminant is D=0D=0.

D= (2mn)2−4(1+m2)(n2−r2)= 4m2n2−4(n2−r2+m2n2−m2r2)= 4m2n2−4n2+4r2−4m2n2+4m2r2= −4(n2−r2−m2r2)=0 \begin{align*} D =&\ (2mn)^2-4(1+m^2)(n^2-r^2) \\ =&\ 4m^2n^2-4(n^2-r^2+m^2n^2-m^2r^2) \\ =&\ 4m^2n^2-4n^2+4r^2-4m^2n^2+4m^2r^2 \\ =&\ -4(n^2-r^2-m^2r^2)=0 \end{align*}

Therefore,

n2= r2m2+r2=r2(m2+1)  ⟹  n= ±rm2+1 \begin{align*} && n^2 =&\ r^2m^2+r^2=r^2(m^2+1) \\ \implies && n =&\ \pm r\sqrt{m^2+1} \end{align*}

Hence, the equation of the tangent line to circle x2+y2=r2x^2+y^2=r^2 with slope mm is

y=mx±rm2+1 y=mx\pm r\sqrt{m^2+1}

■