The value of ' \(k\) ' so that the line \(y=2 x+k\) may touch the ellipse \(3 x^2+5 y^2=15\) is
The value of ' \(k\) ' so that the line \(y=2 x+k\) may touch the ellipse \(3 x^2+5 y^2=15\) is
\(\pm \sqrt{23}\)
\(\pm \sqrt{13}\)
\(\pm \sqrt{33}\)
\(\pm \sqrt{32}\)
Solution
For \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\) and \(y=m x+c\) is tangent Condition of tangency is \(\left(c^2=a^2 m^2+b^2\right)\)
Here, \(\frac{x^2}{5}+\frac{y^2}{3}=1\) and \(y=2 x+k\) is a tangent
\(\begin{array}{ll}
\Rightarrow & k^2=5 \times 4+3 \\
\Rightarrow & k^2=23 \Rightarrow k= \pm \sqrt{23}
\end{array}\)