A Parabola Problem

Number Theory Level 2

Given a parabola ( P ) : y = x 2 (P): y=x^{2} and a straight line ( d 1 ) : y = 2 − x (d_{1}): y = 2 - x , where do ( P ) (P) and ( d 1 ) (d_{1}) intersect?

There are 2 intersections, ( x 1 , y 1 ) (x_{1}, y_{1}) and ( x 2 , y 2 ) (x_{2}, y_{2}) . Type your answer as y 1 + y 2 x 1 + x 2 \dfrac{y_{1}+y_{2}}{x_{1}+x_{2}} . If your answer is not an integer, then do this:

  • If your answer is a rational number (except integers) in the form the ratio between 2 coprime integers, type the sum of those 2 coprime integers.
  • If your answer is an irrational number, type your answer as 0.

Bonus Answer the first question.


The answer is -5.

Tin Le by Tin Le , 15, Vietnam

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1 solution

Δrchish Ray
Jun 5, 2019

To find the two intersection points, we can equate the equations to each other. As a result, we get x 2 = 2 − x x^{2}=2-x

Solving: x 2 + x − 2 = 0 x^2+x-2=0

( x + 2 ) ( x − 1 ) = 0 (x+2)(x-1)=0

x 1 = − 2 \textcolor{#D61F06}{x_1=-2} and x 2 = 1 \textcolor{#D61F06}{x_2=1}

Plugging this back into the second equation, y = 2 − x y=2-x , we get

y 1 = 4 \textcolor{#D61F06}{y_1=4} and y 2 = 1 \textcolor{#D61F06}{y_2=1}

Therefore, y 1 + y 2 x 1 + x 2 = 4 + 1 − 2 + 1 = 5 − 1 = − 5 \frac{y_1+y_2}{x_1+x_2}=\frac{4+1}{-2+1}=\frac{5}{-1}=\boxed{-5}

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