# Find the equation of the tangent to the hyperbola $$x^2 – 4y^2$$ = 36 which is perpendicular to the line x – y + 4 = 0

## Solution :

Let m be the slope of the tangent, since the tangent is perpendicular to the line x – y = 0

$$\therefore$$  m$$\times$$1 = -1 $$\implies$$ m = -1

Since $$x^2-4y^2$$ = 36 or $$x^2\over 36$$ – $$y^2\over 9$$ = 1

Comparing this with $$x^2\over a^2$$ – $$y^2\over b^2$$ = 1

$$\therefore$$; $$a^2$$ = 36 and $$b^2$$ = 9

So the equation of the tangent are y = -1x $$\pm$$ $$\sqrt{36\times {-1}^2 – 9}$$

$$\implies$$ y = x $$\pm$$ $$\sqrt{27}$$ $$\implies$$ x + y $$\pm$$ 3$$\sqrt{3}$$ = 0

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