The equation of the tangent to the curve $x^{\frac{5}{2}}+y^{\frac{5}{2}}=33$ at the point (1, 4) is:
Answer & explanation
Correct answer: option 1
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: $y-4=-\frac{1}{8}(x-1)$