Area of the triangle formed by the lines x -y | vedclass

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Question

Any tangent to the hyperbola at

$\mathrm{P}(\mathrm{a} \sec 	heta, \mathrm{a} 	an 	heta)$ is

$x \sec 	heta-y 	an 	heta=a$    &nb

Vedclass · Accepted Answer

$a^2$

Vedclass · Answer

Any tangent to the hyperbola at

$\mathrm{P}(\mathrm{a} \sec 	heta, \mathrm{a} 	an 	heta)$ is

$x \sec 	heta-y 	an 	heta=a$          .....$(1)$

Also; $\quad \mathrm{x}-\mathrm{y}=0$        ......$(2)$

$\mathrm{x}+\mathrm{y}=0$         .....$(3)$

from $(1),(2) $ and $(3) ;$ we get

$A\left(\frac{a}{\sec 	heta-	an 	heta}, \frac{a}{\sec 	heta-	an 	heta}ight) ; B\left(\frac{a}{\sec 	heta+	an 	heta}, \frac{-a}{\sec 	heta+	an 	heta}ight)$

${\&\,\, \mathrm{C}(0,0)} $

${	ext { Area of } \Delta \mathrm{ABC}=\frac{1}{2}\left(\mathrm{x}_{1} \mathrm{y}_{2}-\mathrm{x}_{2} \mathrm{y}_{1}ight)}$

$={\frac{\mathrm{a}^{2}}{2}\left(\frac{-1}{\sec ^{2} 	heta-	an ^{2} 	heta}-\frac{1}{\sec ^{2} 	heta-	an ^{2} 	heta}ight)} $

${=\frac{\mathrm{a}^{2}}{2}(-2)=-\mathrm{a}^{2}=\mathrm{a}^{2}}$

Area of the triangle formed by the lines $x -y = 0, x + y = 0$ and any tangent to the hyperbola $x^2 -y^2 = a^2$ is :-

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