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If $c \in R$ be such that the line $4 x-y+c=0$ touches the ellipse $x^2+4 y^2=4$, then an equation having all such values of $c$ among its roots is
Options:
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Verified Answer
The correct answer is:
$x^3-x^2-17 x+17=0$
If line $4 x-y+c=0$ touches the ellipse $\frac{x^2}{4}+\frac{y^2}{1}=1$, then
$$
\begin{aligned}
& c^2=4(4)+1 \Rightarrow c^2=17 \\
& \Rightarrow c= \pm \sqrt{17}
\end{aligned}
$$
Now, from the options, the equation
$$
\begin{aligned}
& x^3-x^2-17 x+17=0 \\
\Rightarrow & x^2(x-1)-17(x-1)=0 \\
\Rightarrow \quad & (x-1)\left(x^2-17\right)=0 \\
\Rightarrow \quad & x=1, \pm \sqrt{17}
\end{aligned}
$$
$\because$ The cubic equation $x^3-x^2-17 x+17=0$ have roots $\pm \sqrt{17}$.
Hence, option (c) is correct.
$$
\begin{aligned}
& c^2=4(4)+1 \Rightarrow c^2=17 \\
& \Rightarrow c= \pm \sqrt{17}
\end{aligned}
$$
Now, from the options, the equation
$$
\begin{aligned}
& x^3-x^2-17 x+17=0 \\
\Rightarrow & x^2(x-1)-17(x-1)=0 \\
\Rightarrow \quad & (x-1)\left(x^2-17\right)=0 \\
\Rightarrow \quad & x=1, \pm \sqrt{17}
\end{aligned}
$$
$\because$ The cubic equation $x^3-x^2-17 x+17=0$ have roots $\pm \sqrt{17}$.
Hence, option (c) is correct.
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