Parabola Question - None

Question

Points A and B lie on the parabola $y = 2x^2 + 4x - 2$, such that origin is the mid-point of the line segment AB. If $l$ be the length of the line segment AB, then find the unit digit of $l^2$.

Accepted Answer

Let $A(x_1, y_1)$ and $B(x_2, y_2)$ be points on the parabola $y = 2x^2 + 4x - 2$.

Since origin is the midpoint of AB:

$\frac{x_1 + x_2}{2} = 0 \Rightarrow x_1 + x_2 = 0$

$\frac{y_1 + y_2}{2} = 0 \Rightarrow y_1 + y_2 = 0$

Since both points lie on the parabola:

$y_1 = 2x_1^2 + 4x_1 - 2$ and $y_2 = 2x_2^2 + 4x_2 - 2$

$y_1 + y_2 = 2(x_1^2 + x_2^2) + 4(x_1 + x_2) - 4 = 0$

$2(x_1^2 + x_2^2) - 4 = 0 \Rightarrow x_1^2 + x_2^2 = 2$

From $x_1 + x_2 = 0$: $x_2 = -x_1$

$x_1^2 + x_1^2 = 2 \Rightarrow x_1 = 1, x_2 = -1$ (or vice versa)

$y_1 = 2(1) + 4(1) - 2 = 4$ and $y_2 = 2(1) + 4(-1) - 2 = -4$

$l^2 = (x_1 - x_2)^2 + (y_1 - y_2)^2 = (2)^2 + (8)^2 = 4 + 64 = 68$

Unit digit of $l^2 = 68$ is $\boxed{8}$.

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