(A) Let the point on the $X$-axis be $S = (x, 0)$ and the given point be $P = (3, 4)$.The distance between $S$ and $P$ is $d$, so $d^2 = (x - 3)^2 + (

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(A) Let the point on the $X$-axis be $S = (x, 0)$ and the given point be $P = (3, 4)$.The distance between $S$ and $P$ is $d$, so $d^2 = (x - 3)^2 + (0 - 4)^2$.$d^2 = (x - 3)^2 + 16$.$(x - 3)^2 = d^2 - 16$.If $d Since the square of a real number cannot be negative, there is no real value of $x$ for $d Therefore, $S$ is an empty set if $d < 4$.

Class 11 Mathematics 10-1.Circle and System of Circles Locus Related Problem

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Let $S$ be the set of points on the $X$-axis lying at a distance of $d$ units from $(3, 4)$. Which of the following is true?

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A
$S$ is an empty set if $d < 4$
B
$S$ contains infinitely many points if $d < 4$
C
$S$ contains at least two points if $d = 4$
D
$S$ contains exactly three points for any $d > 4$

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