If the straight line x cos alpha + y sin alph | vedclass

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(B) The equation of the line is $x \cos \alpha + y \sin \alpha = p$.This can be rewritten as $y = -(\cot \alpha) x + p \csc \alpha$.For a line $y = mx

Vedclass · Accepted Answer

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

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(B) The equation of the line is $x \cos \alpha + y \sin \alpha = p$.This can be rewritten as $y = -(\cot \alpha) x + p \csc \alpha$.For a line $y = mx + c$ to be a tangent to the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$,the condition is $c^2 = a^2 m^2 - b^2$.Here,$m = -\cot \alpha$ and $c = p \csc \alpha$.Substituting these values into the condition:$(p \csc \alpha)^2 = a^2 (-\cot \alpha)^2 - b^2$$p^2 \csc^2 \alpha = a^2 \cot^2 \alpha - b^2$Multiplying both sides by $\sin^2 \alpha$:$p^2 = a^2 \cos^2 \alpha - b^2 \sin^2 \alpha$.

If the straight line $x \cos \alpha + y \sin \alpha = p$ is a tangent to the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$,then:

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