If the tangent to the curve 2 y^3=a x^2+x^3 at the point (a,...

If the tangent to the curve $2 y^{3}=a x^{2}+x^{3}$ at the point $(a, a)$ cuts off intercepts $\alpha$ and $\beta$ on the coordinate axes such that $\alpha^{2}+\beta^{2}=61$, then $a=$
(A) $\pm 30$
(B) $\pm 5$
(C) $\pm 6$
(D) $\pm 61$

Key Concepts

Intercepts of Lines

The intercepts of a line are the points where the line crosses the coordinate axes. The x-intercept is found by setting y = 0 in the line’s equation, and the y-intercept is found by setting x = 0. These intercepts can be used in geometric relations or constraints involving the line.

Algebraic Relations Involving Intercepts

Algebraic relations involving intercepts allow the formation of equations that connect the geometric properties of a line with numerical conditions. By expressing the intercepts in terms of the line’s parameters and then applying given conditions (such as the sum of squares of the intercepts equaling a specific value), one can solve for unknown quantities in the problem.

Implicit Differentiation

Implicit differentiation is a method used to differentiate equations in which y is not isolated on one side. This technique involves differentiating both sides of the equation with respect to x while treating y as an implicit function of x, and then solving for dy/dx. It is especially useful when the relationship between x and y is given in a non-explicit form.

Tangent Lines to Curves

The tangent line to a curve at a given point represents the instantaneous direction or slope of the curve at that point. Calculating the tangent involves determining the slope (derivative) at the specific point on the curve, and then using the point-slope form to create the equation of the line, which approximates the curve near the point of tangency.

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