When does the line x cos alpha + y sin alpha | vedclass

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(A) The condition for the line $y = mx + c$ to be a tangent to the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ is $c^2 = a^2 m^2 + b^2$. Given the

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$p^2 = a^2 \cos^2 \alpha + b^2 \sin^2 \alpha$

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(A) The condition for the line $y = mx + c$ to be a tangent to the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ is $c^2 = a^2 m^2 + b^2$. Given the line $x \cos \alpha + y \sin \alpha = p$,we can rewrite it as $y \sin \alpha = -x \cos \alpha + p$,which simplifies to $y = -(\cot \alpha) x + \frac{p}{\sin \alpha}$. Here,$m = -\cot \alpha$ and $c = \frac{p}{\sin \alpha}$. Substituting these into the condition $c^2 = a^2 m^2 + b^2$: $(\frac{p}{\sin \alpha})^2 = a^2 (-\cot \alpha)^2 + b^2$. $\frac{p^2}{\sin^2 \alpha} = a^2 \frac{\cos^2 \alpha}{\sin^2 \alpha} + b^2$. Multiplying both sides by $\sin^2 \alpha$,we get $p^2 = a^2 \cos^2 \alpha + b^2 \sin^2 \alpha$.

When does the line $x \cos \alpha + y \sin \alpha = p$ touch the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$?

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