Find the y-intercept of the tangent line to the curve y=sqrt(x^2+33).

The answer is 33/7.

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Find the y-intercept of the tangent line to the curve y=sqrt(x^2+33). The answer is 33/7.

You neglected to tell us what point on the curve the tangent line intersects. Since you did give us the y-intercept of the tangent line, let's find the point(s).

y=sqrt(x^2+33)

y' = x/sqrt(x^2+33) = x/y

Assume the point(s) on the curve where the tangent line(s) intersect are (p,q).

q= y(p) = sqrt(p^2+33)

Equation of tangent line(s) is: y - q = (p/q) * (x - p)

Or: y = (p/q) * x + q - (p/q) * p

The y-intercept = q - p^2/q = 33/7

(q^2 - p^2)/q = 33/7

(p^2 + 33 - p^2)/sqrt(p^2+33) = 33/7

Numerators are the same so denominators are equal:

sqrt(p^2+33) = 7

Square both sides:

p^2 + 33 = 49

p^2 = 16

p = ±4

q = sqrt((±4)^2+33) = 7

So tangent lines at either of the two points (±4,7) have the same y-intercept = 33/7.

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