Video: CBSE Class X • Pack 4 • 2015 • Question 14

CBSE Class X • Pack 4 • 2015 • Question 14

03:18

Video Transcript

If the coordinates of points 𝐴 and 𝐵 are negative two, negative two and two, negative four, respectively, and point 𝑃 lies on the line segment 𝐴𝐵 such that 𝐴𝑃 is equal to three-sevenths of 𝐴𝐵, find the coordinates of 𝑃.

Let’s first consider the line 𝐴𝐵, where 𝐴 has coordinates negative two, negative two and 𝐵 has coordinates two, negative four. We’re told that 𝐴𝑃 is three-sevenths of the length of 𝐴𝐵. Therefore, point 𝑃 lies three-sevenths of the way along the line. This means that length 𝑃𝐵 must be four-sevenths of the length 𝐴𝑃. This means that the ratio of 𝐴𝑃 to 𝑃𝐵 must be three to four.

We know that if point 𝑃 lies on the line segment 𝐴𝐵, where 𝐴𝑃 and 𝑃𝐵 have ratio 𝑚 to 𝑛, then the coordinates of point 𝑃 will be given by the following. The 𝑥-coordinate will be 𝑛 multiplied by 𝑥 one plus 𝑚 multiplied by 𝑥 two divided by 𝑚 plus 𝑛. The 𝑦-coordinate will be given by 𝑛 multiplied by 𝑦 one plus 𝑚 multiplied by 𝑦 two divided by 𝑚 plus 𝑛, where point 𝐴 has coordinates 𝑥 one, 𝑦 one and point 𝐵 has coordinates 𝑥 two, 𝑦 two. In our case, 𝑥 one, 𝑦 one is negative two, negative two and 𝑥 two, 𝑦 two is two, negative four. The ratios 𝑚 and 𝑛 are three and four, respectively.

The 𝑥-coordinate of point 𝑃 is therefore equal to four multiplied by negative two plus three multiplied by two all divided by three plus four. Four multiplied by negative two is negative eight. Three multiplied by two is equal to six, and three plus four is equal to seven. We’re therefore left with negative eight plus six divided by seven. This is equal to negative two-sevenths. The 𝑥-coordinate of 𝑃 is negative two-sevenths.

The 𝑦-coordinate can be calculated in the same way: four multiplied by negative two plus three multiplied by negative four divided by three plus four. Four multiplied by negative two is negative eight, and three multiplied by negative four is negative 12. Three plus four is equal to seven. Negative eight minus 12 divided by seven is equal to negative twenty sevenths. Therefore, the 𝑦-coordinate of 𝑃 is negative twenty sevenths.

The coordinates of the point 𝑃 that lies on the line segment 𝐴𝐵 are negative two-sevenths, negative twenty sevenths.

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