The area of the figure bounded by y = x2 – 2x + 3 and the line tangent to it at M (2, 3) and y-axis is 8/k, where k is ____

The problem is asking for the area of the figure bounded by the parabola y = x^2 - 2x + 3, the line tangent to it at the point M(2,3), and the y-axis.

Step 1: Find the equation of the tangent line at M(2,3).

The derivative of y = x^2 - 2x + 3 is y' = 2x - 2.

At the point M(2,3), the slope of the tangent line is y'(2) = 2*2 - 2 = 2.

So, the equation of the tangent line is y - 3 = 2(x - 2), or y = 2x - 1.

Step 2: Find the x-intercept of the tangent line.

Set y = 0 in the equation of the tangent line to get 0 = 2x - 1, or x = 1/2.

Step 3: Find the area of the figure.

The figure is a trapezoid with bases 3 (the y-coordinate of M) and 3 - 1/2 (the y-coordinate of the x-intercept of the tangent line) and height 2 - 1/2 (the x-coordinate of M minus the x-intercept of the tangent line).

So, the area of the figure is 1/2 * (3 + 3 - 1/2) * (2 - 1/2) = 8/2 = 4.

Step 4: Set the area equal to 8/k and solve for k.

4 = 8/k

k = 8/4 = 2.

So, the value of k is 2.