smc-1196-48-Spheres

Find the equation of the vector line `underset~v` that passes through `Atext{(5, 2, 3)}` and `B(7, 6, 1)`. (1 mark)

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A sphere has centre `underset~c` at `text{(2, 3, 5)}` and a radius of `5sqrt2` units.
Find the points where the vector line `underset~v` meets the sphere. (3 marks)

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i. `overset(->)(BA) = ((7 – 5),(6 – 2),(1 – 3)) = ((2),(4),(−2)) = 2((1),(2),(−1))`

`underset~v = ((5),(2),(3)) + lambda((1),(2),(−1))`

ii. `text(General point)\ underset~v:`

`x = 5 + lambda`

`y = 2 + 2lambda`

`z = 3 – lambda`

`text(Equation of sphere,)\ underset~c = (2, 3, 5),\ text(radius)\ 5sqrt2:`

`(x – 2)^2 + (y – 3)^2 + (z -5)^2`	`= (5sqrt2)^2`
`(lambda + 3)^2 + (2lambda – 1)^2 + (−lambda – 2)^2`	`= 50`
`lambda^2 + 6lambda + 9 + 4lambda^2 – 4lambda + 1 + lambda^2 + 4lambda + 4`	`= 50`
`6lambda^2 + 6lambda + 14`	`= 50`
`6lambda^2 + 6lambda – 36`	`= 0`
`6(lambda + 3)(lambda – 2)`	`= 0`
`lambda`	`= –3\ text(or)\ 2`

`text(When)\ \ lambda = –3,`

`text(Intersection) = ((5),(2),(3)) – 3((1),(2),(−1)) = ((2),(–4),(6))`

`text(When)\ \ lambda = 2,`

`text(Intersection) = ((5),(2),(3)) + 2((1),(2),(−1)) = ((7),(6),(1))`

A sphere is represented by the equation

`x^2 - 4x + y^2 + 8y + z^2 - 3z + 2 = 0`

Determine the centre `underset~c` and radius of the sphere. (2 marks)

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Find the vector equation of the sphere. (1 mark)

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i. `x^2 – 4x + y^2 + 8y + z^2 – 3z + 2 = 0`

`(x-2)^2 + (y+4)^2 + (z-{3}/{2})^2 + 2 – (89)/(4) = 0`

`(x-2)^2 + (y+4)^2 + (z-{3}/{2})^2 = (81)/(4)`

`:. \ underset~c = ((2),(-4),({3}/{2})) \ , \ text(radius) = (9)/(2)`

ii. `text(Vector equation:)`

`| \ underset~r – ((2),(-4),({3}/{2})) | = (9)/(2)`

Vectors, EXT2 V1 SM-Bank 21