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could someone please explain "family of curves" and give me an example. thx

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I'm not really sure that is "defined" in any detailed way. I suppose it means something like curves derived from a common source exhibiting some type of shared behaviour. Contours? Conics? Spirals? That sort of thing.
One family of curves is the set of hyperbolas xy = a. Another family of curves is the set of hyperbolas x^2 - y^2 = b, where a and b run through the real numbers.
True but you can take all the conic curves together as one family.

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Other answers:

Family curves, eg, hyperbola; f(x)=1/x The family of curves are different intervals of 1/x A curve at 1, a curve at 2, etc Picture a contour map.
If you select a set of values for a and the same set of values for b and graph those members from each family on the same coordinate axes, you will see that each member of one family intersects orthogonally to every member of the other family.

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