Question

The area of the triangle formed by x− and y− intercepts of the parabola y=0.5(x−3)(x+k) is equal to 1.5 square units. Find all possible values of k.

Answer 1

Answer:

k = -2 , k = -1 , k = 0.56 , k = -3.56

Step-by-step explanation:

First find the x and y intercepts of the triangle formed :

To find x - intercept, setting y = 0

$\implies 0 = 0.5\times (x-3)\cdot (x+k)\\\implies x=3\text{ or }x=-k$

Now to find y - intercept, setting x = 0

$\implies y=0.5\times (0-3)\cdot (0+k)\\\\\implies y=-1.5\cdot k$

Now, base = 3 + k, height = y = -1.5·k and Area = 1.5 square units

$\bf\text{Area of triangle = }\frac{1}{2}\times base\times height\\\\1.5=\frac{1}{2}\times (3+k)\times (-1.5\cdot k)\\\\\text{On solving the above equation,}\\\\\implies k=-2\text{ or }k=-1$

Now, to find y-intercepts which lies right to the point x = 3

$\bf\text{Area of triangle = }\frac{1}{2}\times base\times height\\\\1.5=\frac{1}{2}\times (-3-k)\times (-1.5\cdot k)\\\\\text{On solving the above equation,}\\\\\implies k=0.56\text{ or }k=-3.56$

So, k has four possible values ⇒ k = -2 , k = -1 , k = 0.56 , k = -3.56

Answer 2

Answer:

The possible values of k are -0.56, -3.56, -2 and -1.

Step-by-step explanation:

We have the equation of the parabola, $y=0.5(x-3)(x+k)$.

Substituting x=0, we get that $y=0.5\times (-3)\times k$ i.e. y = -1.5k

So, the y-intercept is (0,-1.5k).

Thus, the height of the triangle becomes |-1.5k| = 1.5k

Again, substituting y=0, $0=0.5(x-3)(x+k)$ i.e. $0=(x-3)(x+k)$ i.e. x=3 and x=-k

So, the x-intercepts are (3,0) and (-k,0).

Thus, the base of the triangle is 3+k if k>-3 or -3-k if k<-3.

We see that 'k' cannot be 0 or -3 as if k=0, then height = 0, which is not possible. If k= -3, then the base = 0, which is also not possible.

As, area of a triangle = $\frac{1}{2}\times (base)\times (height)$.

Substituting the values, we get,

1.5=$\frac{1}{2}\times (3+k)\times (1.5k)$

i.e. $3=4.5k+1.5k^2$

i.e. k = -0.56 and k = -3.56

or

1.5=$\frac{1}{2}\times (-3-k)\times (1.5k)$

i.e. $3=-4.5k-1.5k^2$

i.e. k = -2 and k = -1.

Thus, the possible values of k are -0.56, -3.56, -2 and -1.