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A line x –y + 4 = 0 intersects the line segment joining the points A(2,3) and B(-2,4) at point P. What ratio does the line divide AB into? Also find the coordinates of P.

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Final answer:

The line x - y + 4 = 0 intersects line segment AB (points A(2,3) and B(-2,4)) at P. Substituting A and B coordinates, the ratio is 3: -2. This divides AB into segments AP and PB with lengths in the ratio 3: -2. Using the coordinate formula, P's coordinates are (0, 1).

Step-by-step explanation:

The line equation x - y + 4 = 0 intersects the line segment AB joining points A(2,3) and B(-2,4) at point P.

To find the ratio, we'll substitute A and B coordinates into the line equation, obtaining the values of x and y.

For A, x - y + 4 = 0 becomes 2 - 3 + 4 = 3, and for B, it becomes -2 - 4 + 4 = -2. So, the ratio is 3: -2.

The line divides AB into segments AP and PB with lengths proportional to the ratio 3: -2.

To find the coordinates of P, use the formula (x, y) = [(x₁m + x₂n)/(m + n), (y₁m + y₂n)/(m + n)].

Substituting, P's coordinates are (0, 1).

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