Find the co-ordinates of a point which is at a distance of 5 units from the point (7, 9) and 1 unit from the point (3, 5)?

Question

Find the co-ordinates of a point which is at a distance of 5 units from the point (7, 9) and 1 unit from the point (3, 5)?

(4, 5)
(3, 6)
Both of these
None of these

LohaHridoy

6 views

2025-01-04 04:42

1 Answers

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AhmedNazir · Accepted Answer · 2022-09-04T04:29:57+06:00

Option 3 : Both of these

Let A (7, 9) and B(3, 5) be the given points. Let P(x, y) be the required point

⇒ PA = 5 and PB = 1

⇒ PA² = 25 and PB² = 1

Consider P(x, y) and A(7, 9)

⇒ (x - 7)² + (y - 9)² = 25

⇒ x² - 14x + 49 + y² - 18y + 81 = 25

⇒ x² + y² - 14x - 18y + 105 = 0 ...(1)

Consider P(x, y) and B(3, 5)

⇒ (x - 3)² + (y - 5)² = 1

⇒ x² - 6x + 9 + y² - 10y + 25 = 1

⇒ x² + y² - 6x - 10y + 33 = 0 ...(2)

Subtracting equation (2) from (1)

⇒ - 8x - 8y + 72 = 0

⇒ x + y = 9

⇒ y = 9 - x

Substituting value of ‘y’ in equation (2)

⇒ x² + (9 - x)² - 6x - 10(9 - x) + 33 = 0

⇒ x² + 81 - 18x + x² - 6x - 90 + 10x + 33 = 0

⇒ 2x² - 14x + 24 = 0

⇒ x² - 7x + 12 = 0

⇒ (x - 4) (x - 3) = 0

⇒ x = 3 or 4

⇒ If x = 3, y = 9 - x = 9 - 3 = 6

⇒ If x = 4, y = 9 - x = 9 - 4 = 5

∴ required points are (3, 6) and (4, 5)