#### To determine

**To find:**

The downward opening parabola that passes through the point (a, b) and the origin such that the area under the parabola is a minimum.

#### Answer

**y=-2ba2x2+3bax**

#### Explanation

**1) Concept:**

Substitute the point a, b in the equation of parabola, and find the value of q in terms of p.

**2) Calculation:**

It is given that the parabola is downward opening, so the coeffient of x2 is negative.

Let the equation of the parabolabe y=-px2+qx+r , p>0

Given that parabola passes through point (0, 0).

So substitute (0, 0) in the equation of parabola.

0=-p02+q(0)+r

Simplify.

r=0

Substitute r=0 in the equation of parabola.

y=-px2+qx+0

Now the equation of parabola is

y=-px2+qx

Given that parabola passes through point a, b.

Substitute (a, b) in the equation of parabola, and find the value of q in terms of p.

b=-pa2+qa

Add pa2 to both the sides.

b+pa2=-pa2+qa+pa2

Simplify.

b+pa2=qa

Divide by a, and simplify.

q=b+pa2a……….say(i)

For getting the intersection points of X- axis and parabola,

Put y=0 in the equation of parabola.

0=-px2+qx

Factor out x, and rearrange.

x(-px+q)=0

That is,x=0, x=qp

Now the area under the parabola is given by the area between x- axis and parabola.

A=∫0qp(-px2+qx)dx

Solve the above the integral to get,

A=-px33+qx220qp

Substitute limits.

A=-p(qp)33+q(qp)22

Simplify.

A=q36p2

Substitute q=b+pa2a in the above equation.

A=b+pa2a36p2

Simplify.

A=b+pa236p2a3

It is given that the area under the parabola is minimum, therefore it is zero at its derivative.

dAdp=6p2a3·ddpb+pa23-b+pa23·ddp(6p2a3)6p2a32

dAdp=6pa3·b+pa223pa2-2b-2pa26p2a32

Equate dAdp=0

b+pa22pa2-2b6p3a3=0

Since area is non-zero b+pa2 is non-zero. Thus we have

pa2-2b=0

Solving for p we have

p=2ba2 Put the above equation in q=b+pa2a

q=b+(2ba2)a2a

Simplify.

q=3ba

Put the above values of p, q in the equation of parabola y=-px2+qx

y=-(2ba2)x2+(3ba)x

Therefore, the equation of parabola is y=-2ba2x2+3bax

**Conclusion:**

The equation of the parabola is y=-2ba2x2+3bax