Question

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

• A. x + 8y − 33 = 0
• B. 12x + y − 8 = 0
• C. x + 8y − 12 = 0
• D. x + 12y − 8 = 0

Answer 1

The Correct Option is A

Differentiate the given curve implicitly:

\[\frac{d}{dx} \left( \frac{5}{x^2} + \frac{5}{y^2} \right) = \frac{d}{dx}(33).\]

Using the chain rule:

\[-\frac{10}{x^3} - \frac{10}{y^3} \cdot \frac{dy}{dx} = 0.\]

Rearrange to find \(\frac{dy}{dx}\):

\[\frac{dy}{dx} = -\frac{10}{x^3} \div -\frac{10}{y^3} = -\frac{y^3}{x^3}.\]

At the point (1, 4):

\[\frac{dy}{dx} = -\frac{4^3}{1^3} = -64.\]

The equation of the tangent is:

\[y - y_1 = m(x - x_1),\]

where \(m = -64\), \((x_1, y_1) = (1, 4)\). Substituting:

\[y - 4 = -64(x - 1).\]

Simplify:

\[y - 4 = -64x + 64 \quad \Rightarrow \quad x + 8y - 33 = 0.\]