We have $x = 11-13i$ and $y = 35-i$.
$a$ is a complex number which trisects the line segment joining $x$ and $y$. $a$ is also closer to $x$ than $y$. Find $a$.
I'm not sure where to start. Would a diagram help? Hints only please.
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By "trisects" you mean in three segments of equal length? Not sure about the names in english. – Timbuc Apr 14 '15 at 17:53
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@Timbuc: I believe that's what the problem means, yes. Trisects = three segments of equal length. – Math is Life Apr 14 '15 at 17:54
2 Answers
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Hint:
The line segment defined by points $\;A(x_1,y_1)\,,\,\,B(x_2,y_2)\;$ in the plane gets divided by point $\;P(x_p,y_p)\;$ on the line in the ratio $\;\dfrac{AP}{PB}=\dfrac{k_1}{k_2}\;$ iff
$$x_p=\frac{k_1x_2+k_2x_1}{k_1+k_2}\;,\;\;\;y_p=\frac{k_1y_2+k_2y_1}{k_1+k_2}$$
Spoiler !
Your ratio is $\;xa:ay=1:2\;$
Timbuc
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Let a be the point on the line joining x and y and 1/3 the distance from x to a as from x to y. the horizontal distance from x to y is 35 - 11 = 24,the change in x coordinates.the point a is 1/3 the distance along this line parallel to the x axis,so the x c00rdinate of a must be the x coordinate of x + (1/3)*the distance from x to y.or a(x coordinate = 11 + (1/3)(35 - 11) = 11 + 8 = 19. Similarly,the y coordinate of x = -13, and the y coordinate of y is -1,a difference of 12.Therefore, the y coordinate of a must be (1/3) of -1 -(-13) =12 or 4 units from the y coordinate of x toward the x axis.So the y coordinate of a is -13 + 4 = -9.Combining these results,the coordinates of a are 19 - 9i. Edwin Gray
Edwin Gray
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1(1) Try to learn the easy directions to properly write mathematics in this site, that way more people will read your posts: http://meta.math.stackexchange.com/questions/5020/mathjax-basic-tutorial-and-qu%E2%80%8C%E2%80%8Bick-reference (2) Do not sign your posts: we all can see the name of every poster. – Timbuc Apr 15 '15 at 12:09