Question

The area of triangle formed by points of intersection of parabola y=a(x−1)(x−4) with the coordinate axes is 3. Find a if it is known that parabola opens downward.

Answer 1

To find the value of 'a', we can find the x-intercepts and y-intercept of the parabola, and then use the formula for the area of a triangle to solve for 'a'.

Step-by-step explanation:

To find the area of the triangle formed by the points of intersection of the parabola with the coordinate axes, we need to find the x-intercepts and y-intercepts of the parabola. The x-intercepts occur when y=0, so we set the equation equal to zero and solve for x. Similarly, the y-intercepts occur when x=0, so we substitute x=0 into the equation to find the y-coordinate.

Given that the area of the triangle is 3, we can use the formula for the area of a triangle, which is half the base multiplied by the height. In this case, the base of the triangle is the difference between the x-intercepts, and the height is the y-intercept.

By substituting the x-intercepts and y-intercept into the area formula and simplifying, we can solve for 'a'.

Learn more about Finding the value of 'a'

Answer 2

The value of a will be
`a=-(1)/(2)`

Step-by-step explanation:

Start by graphing the parabola and the three points of the triangle. These points are at intersections of y=a(x-1)(x-4) and the axes

so the y = 0 points are (1,0) and (4,0).

The x = 0 intersection is when y=a(-1)(-4) or (0,4a)

The base and height of this triangle are

The base would be the distance between the y=0 intersections and the height would be the y value of the other vertex.

Hence, base=3 units and height = 4a units. Thus, area can be calculated as

`A = (1)/(2)* b* h =(1)/(2)* 3* 4a = 3`

`a=(1)/(2)`

∵ the parabola opens downward therefore a will be negative.

hence,
`a=-(1)/(2)`