Question

A man running a racecourse notes that the sum of the distances from the two flag posts form him is always 10 m and the distance between the flag posts is 8 m. find the equation of the posts traced by the man.

Answer 1

Let F₁ and F₂ be two points where the flag posts are fixed on the ground.

The origin O is the mid-point of F₁F₂.

∴ OF₁ - OF₂ = `1/2 F_1F_2 = 1/2 xx 8 = 4m`

Coordinates of F₁ are (-4, 0) and F₂ are (4, 0)

Let P (α, β) be any point on the track.

∴ PF₁ + PF₂ = 10

∴ `sqrt((α + 4)^2 + (β - 0)^2)` + `sqrt((α - 4)^2 + (β - 0)^2) = 10`

= `sqrt(α^2 + 16 + 8α + β^2)` =  `10 - sqrt(α^2 + 16 - 8α + β^2)`

Squaring both sides, we get

α² + β² + 8α + 16 = 100 + α² + β² - Bα + 16 `-20 sqrt(α^2 + β^2 - 8α + 16)`

= 16α - 100 = `-20 sqrt(α^2 + β^2 - 8α + 16)`

Again, squaring both sides, we get

= (16α - 100)² = (`-20 sqrt((α^2 + β^2 - 8α + 16)^2)`

= 256α² + 10000 - 3200α = 400(α² + β² - 8α+ 16)

= 256α² + 10000 - 3200α = 400α² + 400β² - 32008α+ 6400)

= 144α² + 400β² = 3600

= `(144α^2)/(3600) + (400β^2)/(3600) = 1`

Thus, the required equation of locus of point P is `x^2/25 + y^2/9 = 1`.