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I have been having difficulty answering the following question:

The figure below shows a distribution between the origin, O, and the point B. The section from O to A is represented by the equation $y = x^2$, whilst the section from point A to point B is represented by the equation $3y = −20+ 20x −2x^2$. Find the coordinates of the point A.enter image description here

Any help in terms of how to apply the correct methodology for this question and how to solve it would be greatly appreciated.

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    The point A satisfies both equations. – 2017-01-08

2 Answers 2

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Let $ A=(x_0,y_0) $. A is the intersection point between the two curves, which means that $ y_0 = x_0^2 $ and that $ y_0 = \frac{1}{3} \big( -20 + 20x_0 - 2x_0^2 \big) $.

Let $ y_0 = y_0 $, i.e.

$$ x_0^2 = \frac{1}{3} \big( -20 + 20x - 2x^2 \big) $$

$$ 3x_0^2 = -20 + 20x_0 - 2x_0^2 $$

$$ 5x_0^2 - 20x_0 + 20 = 0 $$

$$ 5 (x_0^2 - 4x_0 + 4) = 0 $$

$$ 5(x_0-2)(x_0-2) = 0 $$

$$ x_0 = 2 $$

Plugging this into either of the two equations, you get: $ y_0 = x_0^2 = (2)^2 = 4 $. Therefore, $ A=(x_0,y_0) = (2,4). $

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    just check $x_0$ it should be $2$ – 2017-01-08
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    @Karl you're right, thanks! I've updated my answer. – 2017-01-08
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    Thankyou this makes lots of sense, Did not realise it was simple algebraic manipulation, it looked to me like a vectors question – 2017-01-09
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Plugging $y=x^2$ in the second equation, we get after simplification

$$x^2-4x+4=0$$ which gives the coordinates of the point $A : x=2$ and $y=4$.