HSC 1993 2U Integration Question (1 Viewer)

[RivalryofTroll]

(No diagram available)

In the diagram, A(-1,1) and B(2,4) are the points of intersection of the parabola y=x^2 with the line y=x+2. The point P(t,t^2) is a variable point on the parabola below the line.

(i) Find the area of the parabolic segment APB.

Found it: A = 9/2 units^2

(ii) Show that the maximum area of triangle APB is 3/4 of the area of the parabolic segment APB.

Yeah, hints on how to do part (ii)?

 

[Carrotsticks]

For part (i), how did you get a constant value of the area of triangle APB? P is a variable point and the area of the triangle is not constant. This can be demonstrated by moving P arbitrarily close to A or B, which forces the area to approach zero.

 

[RivalryofTroll]

For part (i), how did you get a constant value of the area of triangle APB? P is a variable point and the area of the triangle is not constant. This can be demonstrated by moving P arbitrarily close to A or B, which forces the area to approach zero.

I got a constant value for the parabolic segment APB NOT the triangle APB (i.e. the area below the line and above the parabola).

 

[RivalryofTroll]

And to confirm, I checked the answers for part (i) to make sure I was correct. (answers unavailable for part ii)

 

[Carrotsticks]

My fault, I mis-read your expression to be 'triangle APB'.

To find the maximum possible area of triangle APB, we must of course find an expression for it. It is bounded by the three points (-1,1), (t,t^2) and (2,4). To find the area, you could shift (-1,1) to become the origin, construct a 2x2 matrix and find the determinant (good way of verifying that your answer is correct).

Otherwise, you will need to use the formula A = 1/2 * base * height.

The 'base' of the triangle is the length AB (easy to find) and the 'height' would be to find the perpendicular distance of P to AB, using the perpendicular distance formula.

Once you find those, you will have Area triangle APB = .... (something in terms of t), then you do the good ol' differentiate, let it equal 0 blah blah etc.