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Let $S$ be the set of points on $X$-axis lying at a distance of $d$ units from $(3,4)$. Which of the following is true?
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The correct answer is:
$S$ is an empty set if $d < 4$
Let $S=(x, 0)$ and $P=(3,4)$
$\therefore \quad d^2=S P^2=(x-3)^2+16 \Rightarrow(x-3)^2=d^2-16$
If $d \in(-4,4)$, then $\left(+d^2-16 < 0\right)$ and then $(x-3)^2$ becomes negative, which is impossible.
$\Rightarrow \quad S$ is an empty set if $d < 4$.
$\therefore \quad d^2=S P^2=(x-3)^2+16 \Rightarrow(x-3)^2=d^2-16$
If $d \in(-4,4)$, then $\left(+d^2-16 < 0\right)$ and then $(x-3)^2$ becomes negative, which is impossible.
$\Rightarrow \quad S$ is an empty set if $d < 4$.
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