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In ℝ2, you are given the points A(−14,15) and X(13,−17). Find t such that the point C(−12,t) lies on the line through A and X. answer as a ratio not as a decimal

Respuesta :

steffaniasierrag

Answer:

The equation of the line segment between points A and X is

[tex]sA + (1-s)X, s\in[0,1][/tex]

Then we need that the point C satisfy:

[tex]\left[\begin{array}{c}-12\\t\end{array}\right] =\left[\begin{array}{c}-14s\\15s\end{array}\right]+\left[\begin{array}{c}13(1-s)\\-17(1-s)\end{array}\right][/tex].

This implies that

[tex]-12=-14s-13s+13\\s=\frac{25}{27}[/tex]

We replace the value of s in the equation we get from the second component

[tex]t=15s+17s-17\\t=32\frac{25}{27}-17\\t=\frac{341}{27}[/tex]

Then the point [tex]C=(-12,\frac{341}{27})[/tex] lies on the line through A and X.

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