Question

1-The tangent line to the curve y = x³ at point A(a, a³) intersects the curve at point B. Let

R be the area of the region bounded by the curve and the tangent line. The tangent line at B
intersects the curve at another point C. Let S be the area of the region bounded by curve and
the second tangent line. How are the areas R and S related?

Answer 1

The areas R and S are equal.

Explanation:

Since, the tangent line to the curve y = x³ at point A(a, a³) is given by y = 3a²(x - a) + a³. This line intersects the curve at point B(a, a³), and we can find the x-coordinate of point B by setting y = 3a²(x - a) + a³ equal to x³ to get the equation x³ - 3a²x + 3a³ - a³ = 0. Using the fact that x = a is a root of this equation, we can factor it as (x - a)³ = 0, which implies that the curve and the tangent line are tangent at point B.

The slope of the tangent line at point B is 3a², so the equation of the tangent line at point B is y = 3a²(x - a) + a³. This line intersects the curve at another point C, and we can find the x-coordinate of point C by setting y = 3a²(x - a) + a³ equal to x³ to get the equation x³ - 3a²x + 3a³ - a³ = 0. Using the fact that x = a is a root of this equation, we can factor it as (x - a)³ = 0, which implies that the curve and the tangent line are tangent at point C.

The region bounded by the curve and the tangent line at point A is a triangle with base a - (a - a) = 0 and height a³ - a³ = 0, so its area is 0. The region bounded by the curve and the tangent line at point B is a triangle with base a - a = 0 and height a³ - a³ = 0, so its area is also 0. Therefore, the areas R and S are both equal to 0, which means they are equal to each other.