The equation of the tangent to the curve $x^{\frac{5}{2}}+y^{\frac{5}{2}}=33$ at the point (1, 4) is:

$x+8 y-33=0$

The correct answer is Option (1) → $x+8 y-33=0$

Given curve:

$x^{\frac{5}{2}}+y^{\frac{5}{2}}=33$

Differentiate implicitly:

$\frac{5}{2}x^{\frac{3}{2}}+\frac{5}{2}y^{\frac{3}{2}}\frac{dy}{dx}=0$

$\frac{dy}{dx}=-\frac{x^{\frac{3}{2}}}{y^{\frac{3}{2}}}$

At point $(1,4)$:

$\frac{dy}{dx}=-\frac{1^{\frac{3}{2}}}{4^{\frac{3}{2}}}=-\frac{1}{8}$

Equation of tangent:

$y-4=-\frac{1}{8}(x-1)$

final answer: $y-4=-\frac{1}{8}(x-1)$