Prove quadratic equation \(\displaystyle{\left({y}={a}{x}^{{{2}}}+{b}{x}+{c}\right)}\) has only one

amantantawq5l

amantantawq5l

Answered question

2022-03-31

Prove quadratic equation (y=ax2+bx+c) has only one line of symmetry

Answer & Explanation

arrebusnaavbz

arrebusnaavbz

Beginner2022-04-01Added 18 answers

Step 1
Symmetry around x=s is expressed by
ax2+bx+c=a(2sx)2+b(2sx)+c
for all x.
By identification of the powers of x, we get the compatibility condition
b+2as=0
or
s=b2a
Step 2
Consider an arbitrary parabola and an arbitrary line. By rotation we can bring the line on the axis y. Then the equation of the rotated parabola
ax2+2bxy+cy2+2dx+2ey+f=0
must be invariant to a change of the sign of x and it must reduce to
ax2+cy2+2ey+f=0.
For this equation to represent a parabola, we must have c=0. Then,
ax2+2ey+f=0
is a parabola for which the axis of symmetry is y. Hence any line which is an axis of symmetry is the standard axis.
Step 3
Note that in the case ac0 (centered conic), by translation we can further reduce to
ax2+cy2+f=0
and the line is one of the known axis of symmetry. Hence there are exactly two of them.

Do you have a similar question?

Recalculate according to your conditions!

Ask your question.
Get an expert answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?