Answer:
Option B
Explanation:
We know that equation of normal of the hyperbola x2a2−y2b2=1 is
a2xx1+b2yy1=a2−b2
∴ Equations of normal to the hyperbola \frac{x^{2}}{9}-\frac{y^{2}}{4}=1
is
\frac{9x}{x_{1}}+\frac{4y}{y_{1}}=9+4
\Rightarrow \frac{9x}{x_{1}}+\frac{4y}{y_{1}}=13
Since, line x+y=k is normal to the given hyperbola
\therefore \frac{\frac{9}{x_{1}}}{1}=\frac{\frac{4}{y_{1}}}{1}=\frac{13}{k}
\Rightarrow \frac{9}{x_{1}}=\frac{4}{y_{1}}
= \frac{13}{k}
\Rightarrow x_{1}= \frac{9k}{13}
y_{1}=\frac{4k}{13}
Since ((x_{1},y_{1}) lie on the hyperbola
\therefore \frac{\left(\frac{9k}{13}\right)^{2}}{9}-\frac{\left(\frac{4k}{13}\right)^{2}}{9}=1
\Rightarrow \frac{9k^{2}}{169}-\frac{4k^{2}}{169}=1
\Rightarrow 5k^{2}=169
\Rightarrow k=\pm\frac{13}{\sqrt{5}}