Question

If the distance of points `2hati + 3hatj + λhatk` from the plane `r.(3hati + 2hatj + 6hatk)` = 13 is 5 units, then λ = ______.

Options

• `6, -17/3`

• `6, 17/3`

• `-6, - 17/3`

• `-6, 17/3`

Answer 1

If the distance of points `2hati + 3hatj + λhatk` from the plane `r.(3hati + 2hatj + 6hatk)` = 13 is 5 units, then λ = `underlinebb(6, -17/3)`.

Explanation:

The coordinates of a given point is (2, 3, λ).

So, equation of the plane is `vecr.(3hati + 2hatj + 6hatk)` = 13

`\implies (xhati + yhatj + zhatk).(3hati + 2hatj + 6hatk)` = 13

`\implies` 3x + 2y + 6z – 13 = 0

Therefore, distance of the plane from the given point (2, 3, λ) will be

`|(3 xx 2 + 2 xx 3 + 6 xx λ - 13)/sqrt(3^2 + 2^2 + 6^2)|` = 5 ...[Given]

`\implies` ± 5 = `(6λ - 1)/sqrt(49)`

`\implies` ± 35 = 6λ – 1

`\implies` 35 = 6λ – 1 or – 35 = 6λ – 1

Hence, λ = `6, -17/3`