Calculate the distance from the origin to the tangent

Calculate the distance from the origin to the tangent to the lemniscate $\rho^{2}=a^{2}\cos2\theta$.

Step 1: Polar distance-to-tangent formula

For a polar curve $\rho=\rho(\theta)$, the perpendicular distance from the pole $O$ to the tangent at $(\rho,\theta)$ is

\[ d = \frac{\rho^{2}}{\sqrt{\rho^{2}+{\rho'}^{2}}}. \tag{1} \]

Step 2: Differentiate the lemniscate

\[ 2\rho\rho' = -2a^{2}\sin2\theta \;\Longrightarrow\; \rho' = -\frac{a^{2}\sin2\theta}{\rho}. \tag{2} \]

Step 3: Simplify $\rho^{2}+{\rho'}^{2}$

Using $\rho^{2}=a^{2}\cos2\theta$, the numerator of $\rho^{2}+{\rho'}^{2}=\dfrac{\rho^{4}+a^{4}\sin^{2}2\theta}{\rho^{2}}$ collapses via $\rho^4=a^4\cos^22\theta$: \[\rho^{4}+a^{4}\sin^{2}2\theta = a^{4}\cos^{2}2\theta+a^{4}\sin^{2}2\theta = a^{4}, \qquad \rho^{2}+{\rho'}^{2} = \frac{a^{4}}{\rho^{2}}. \tag{3}\]

Step 4: Apply (1)

\[ d = \frac{\rho^{2}}{a^{2}/\rho} = \frac{\rho^{3}}{a^{2}} = \frac{a^{3}(\cos2\theta)^{3/2}}{a^{2}} = a(\cos2\theta)^{3/2}. \tag{4} \]

Step 5: Verification

\[\theta=0\text{: } \rho=a,\ \text{tangent is vertical, } d=a;\ \text{formula gives } a\cdot1^{3/2}=a\ \checkmark\]

\[\theta=0.3,\ a=1\text{: } d=\rho^3\approx0.74980,\ a(\cos2\theta)^{3/2}\approx0.74980\ \checkmark\]