Question

Write the equation of the parabola with (5,-8) and (5,-1)

Answer 1

Answer on Question #40076 - Math – Geometry

Assignment

Write the equation of the parabola with (5,-8) and (5,-1)

Solution

It is impossible to solve this problem, if we do not know additional assumptions.

Suppose that it is parabola with vertical axis of symmetry

$$y = a x ^ {2} + b x + c$$

Points (5,-8) and (5,-1) satisfy this equation

$$\begin{array}{l} - 8 = 2 5 a + 5 b + c \\ - 1 = 2 5 a + 5 b + c \\ \end{array}$$

This system has no solution, so consider the second case.

Suppose that it is parabola with horizontal axis of symmetry

$$x = a y ^ {2} + b y + c$$

Points (5,-8) and (5,-1) satisfy this equation

$$\begin{array}{l} 5 = 6 4 a - 8 b + c \\ 5 = a - b + c \\ \end{array}$$

Subtract the second equation from the first one and get $63a - 7b = 0$ , $9a - b = 0$ or $b = 9a$ .

Substitute $b = 9a$ into equations of the system

$$\begin{array}{l} 5 = 6 4 a - 7 2 a + c \\ 5 = a - 9 a + c \\ \end{array}$$

or

$$\begin{array}{l} 5 = - 8 a + c \\ 5 = - 8 a + c \\ \end{array}$$

Finally

$$c = 5 + 8 a, b = 9 a.$$

Thus, we search an equation

$$x = a y ^ {2} + b y + c$$

In the form

$$x = a y ^ {2} + 9 a y + 8 a + 5$$

To find $a, b, c$ , we need more information.