If points $(t, 2t)$, $(-2, 6)$ and $(3, 1)$ are collinear, then $t$ is equal to

Method 1: Using Slopes (Easiest & Fastest Method)

If three points are collinear, then the slope between any two pairs of points must be equal.

Let's call our points:

• Point A: $(t, 2t)$
• Point B: $(-2, 6)$
• Point C: $(3, 1)$

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Find the slope between points B and C:

$\text{Slope of BC} = \frac{1 - 6}{3 - (-2)} = \frac{-5}{5} = -1$

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Find the slope between points A and B:

$\text{Slope of AB} = \frac{6 - 2t}{-2 - t}$

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Set the slopes equal since points are collinear:

$\frac{6 - 2t}{-2 - t} = -1$

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Solve for $t$:

Cross multiply: $6 - 2t = -1(-2 - t)$

$6 - 2t = 2 + t$

$6 - 2 = t + 2t$

$4 = 3t$

$t = \frac{4}{3}$

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Method 2: Using Area Formula (Alternative Method)

If three points are collinear, the area of the triangle they form is zero.

Using the area formula:

$\text{Area} = \frac{1}{2}|x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)|$

Where $(x_1, y_1) = (t, 2t)$, $(x_2, y_2) = (-2, 6)$, $(x_3, y_3) = (3, 1)$

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Substitute into the formula:

$\frac{1}{2}|t(6 - 1) + (-2)(1 - 2t) + 3(2t - 6)| = 0$

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Simplify inside the absolute value:

$\frac{1}{2}|t(5) + (-2)(1 - 2t) + 3(2t - 6)| = 0$

$\frac{1}{2}|5t + (-2 + 4t) + (6t - 18)| = 0$

$\frac{1}{2}|5t - 2 + 4t + 6t - 18| = 0$

$\frac{1}{2}|15t - 20| = 0$

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Since area equals zero:

$|15t - 20| = 0$

The absolute value equals zero only when the expression inside equals zero:

$15t - 20 = 0$

$15t = 20$

$t = \frac{4}{3}$

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Answer: $t = \frac{4}{3}$