Consider the statement: "A point on the curve $y = x²$ is equidistant from the points $(-1,0)$ and $(5,0).$"

A point on the curve $y = x^2$ has coordinates $(x, x^2)$.

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Using the distance formula:

Distance from $(x, x^2)$ to $(-1, 0)$: $\sqrt{(x + 1)^2 + (x^2)^2}$

Distance from $(x, x^2)$ to $(5, 0)$: $\sqrt{(x - 5)^2 + (x^2)^2}$

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Since the point is equidistant from both points:

$\sqrt{(x + 1)^2 + (x^2)^2} = \sqrt{(x - 5)^2 + (x^2)^2}$

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$(x + 1)^2 + (x^2)^2 = (x - 5)^2 + (x^2)^2$

The $(x^2)^2$ terms cancel:

$(x + 1)^2 = (x - 5)^2$

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$x^2 + 2x + 1 = x^2 - 10x + 25$

The $x^2$ terms cancel:

$2x + 1 = -10x + 25$

$12x = 24$

$x = 2$

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Since the point is on $y = x^2$:

$y = (2)^2$

$y = 4$

Therefore, the point $(2, 4)$ on the curve $y = x^2$ is equidistant from $(-1, 0)$ and $(5, 0)$.