Question

In ΔABC the mid-point of the sides AB, BC and CA are respectively (l, 0, 0), (0, m, 0) and (0, 0, n). Then, `("AB"^2 + "BC"^2 + "CA"^2)/("l"^2 + "m"^2 + "n"^2)` is equal to ______.

विकल्प

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• 8

• 16

Answer 1

In ΔABC the mid-point of the sides AB, BC and CA are respectively (l, 0, 0), (0, m, 0) and (0, 0, n). Then, `("AB"^2 + "BC"^2 + "CA"^2)/("l"^2 + "m"^2 + "n"^2)` is equal to 8.

Explanation:

From the figure

x₁ + x₂ = 2l, y₁ + y₂ = 0, z₁ + z₂ = 0

x₂ + x₃ = 0, y₂ + y₃ = 2m, z₁ + z₃ = 0

and x₁ + x₃ = 0, y₁ + y₃ = 0, z₁ + z₃ = 2n

On solving, we get x₁ = l, x₂ = l, x₃ = –1

y₁ = –m, y₂ = m, y₃ = m and z₁ = n, z₂ = –n, z₃ = n

∴ Coordinates are A(l, –m, n), B(l, m, –n) and C(–l, m, n)

∴ `("AB"^2 + "BC"^2 + "CA"^2)/("l"^2 + "m"^2 + "n"^2)`

= `((4"m"^2 + 4"n"^2) + (4"l"^2 + 4"n"^2) + (4"l"^2 + 4"m"^2))/("l"^2 + "m"^2 + "n"^2)`

= 8