The line $21 x+5 y=k$ touches the hyperbola $7 x^2-5 y^2=$ 232 then $k=$
The line $21 x+5 y=k$ touches the hyperbola $7 x^2-5 y^2=$ 232 then $k=$
$116$
$232$
$58$
$110$
Solution
The line $\mathrm{A} x+\mathrm{B} y+\mathrm{C}=0$ touches the hyperbola
$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ if $\mathrm{C}^2=a^2 \mathrm{~A}^2-b^2 \mathrm{~B}^2$ ...(i)
Now, given line is $21 x+5 y=k$ and hyperbola $7 x^2-5 y^2=232 \Rightarrow \frac{x^2}{\left(\frac{232}{7}\right)}-\frac{y^2}{\left(\frac{232}{5}\right)}=1$
So, from (i) we have $k^2=\frac{232}{7}(21)^2-\frac{232}{5}(5)^2 \Rightarrow k=116$