Question

The point on the hyperbola $x^2-9y^2=9$, where the line 5x + 12y = 9 touches it, is .

• A. $(5,-\frac{4}{3})$
• B. $(-5,\frac{4}{3})$
• C. $(-\frac{5}{3},4)$
• D. $(\frac{5}{3},-4)$

Answer 1

The correct option is A $(5,-\frac{4}{3})$

$\mathrm{The}\mathrm{equation}\mathrm{of}\mathrm{the}\mathrm{line}\mathrm{which}\mathrm{touches}\mathrm{the}\mathrm{hyperbola}{x}^2-9y^2=9\mathrm{at}(x_1,y_1)\mathrm{is}{x}_{1}x-9y_{1}y=9\mathrm{or}\frac{x_1}{9}+\left(-y_1\right)y=1\mathrm{Comparing}\mathrm{it}\mathrm{with}\mathrm{the}5x+12y=9\mathrm{or}\frac{5}{9}x+\frac{4}{3}y=1\mathrm{we}\mathrm{get}\frac{x_1}{9}=\frac{5}{9}\Rightarrow x_1=5-y_1=\frac{4}{3}\Rightarrow y_1=-\frac{4}{3}\mathrm{So},\mathrm{the}\mathrm{point}\mathrm{is}\left(5,-\frac{4}{3}\right)$