#### To determine

**To prove:**

The area of parabola segment is 43 times the area of the inscribed triangle ABC.

#### Answer

Result is proved.

#### Explanation

**1) Concept:**

The coordinates of point C: Place the equation of line in the parabola, and the intersecting points will give us C coordinates.

Then find the area of parabola segment =Area under the parabola – Area of triangle ABD(area under the line)

Then calculate the ratio of area of parabola segment ACB to the area of the triangle.

2) **Calculation:**

Let the line parallel to y=x+2 is y=x+k

Since it touches the parabola, it will satisfy the equation of the parabola at a point c.

x+k=4-x2

Add x2-4 on both sides, and simplify.

x2+x+k-4=0

Discriminant D for the above equation should be zero because the line is tangent to the parabola.

D=b2-4ac=0

Here, a=1, b=1, c=k-4

Substitute above values in D.

D=12-41k-4=0

Simplify.

k=174

So y=x+k=x+174

To get the coordinates of point C, substitute the equation of line in the parabola, and the intersecting points will give us the C coordinates.

x+174=4-x2

Add x2-4 to both the sides, and simplify.

x2+x+14=0

Solve the above equation.

x=-12

Substitute the above value in y=x+174.y=-12+174

Simplify.

y=154

Therefore, the coordinates of C are -12, 154. To calculate points of intersection of the given line and parabola,

x+2=4-x2

Add x2-4 on both sides, and simplify.

x2+x-2=0

Simplify.

x=-2, 1

Put x=-2, 1 in y=x+2

y=-2+2=0 And

y=1+2=3

Now the area of the parabola segment = Area under parabola – Area of triangle ABD (area under line). Thus

A=∫-21(4-x2)dx-∫-21(x+2)dx

Integrating.

A=4x-x33-21-x22+2x-21

Take limits.

A=41-133-4-2-(-233-122+21-(-2)22+2-2

Simplify.

A=92

Area of the triangle with coordinates Ax1,y1, Bx2,y2 and Cx3,y3 is given by

a=12x1y11x2y21x3y31

Area of the triangle ACB can be written in determinant form as we know the coordinates.

a=12-201131-11541=278

Now calculate the ratio of area of parabola segment ACB to area of triangle.

Aa=92278=43

Therefore, the area of parabola segment is 43 of the area of triangle ABC.

**Conclusion:**

The area of parabola segment is 43 of the area of triangle ABC.