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प्रश्न
Given A ≡ (x, x + 1) and B ≡ (3, 7). Find x, if AB = 15 units.
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उत्तर
Given:
Points A = (x, x + 1) and B = (3, 7)
Distance (AB = 15 ) units
The distance formula between two points (x1, y1) and (x2, y2)
\[AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]
Substitute points A = (x, x + 1) and B = (3, 7) and distance AB = 15
\[15 = \sqrt{(3 - x)^2 + (7 - (x+1))^2} \]
\[15 = \sqrt{(3 - x)^2 + (7 - x - 1)^2} \]
\[15 = \sqrt{(3 - x)^2 + (6 - x)^2} \] ... [Simplify inside the square root.]
152 = (3 − x)2 + (6 − x)2 ... [Square both sides to remove the square root.]
225 = (3 − x)2 + (6 − x)2
225 = (3 − x)2 + (6 − x)2 = (x − 3)2 + (x − 6)2 ...[Expand the squares]
225 = (x2 − 6x + 9) + (x2 − 12x + 36)
225 = 2x2 − 18x + 45
2x2 − 18x + 45 − 225 = 0
2x2 − 18x − 180 = 0
x2 − 9x − 90 = 0 ... [Simplify by dividing the entire equation by 2.]
Factors of (−90) that add to (−9) are (+6) and (−15)
(x − 15) (x + 6) = 0
x − 15 = 0
x = 15
x + 6 = 0
x = −6
