1a)Find an equation for the surface consisting of all points P for which the distance from P to the y axis is twice the distance from P to the xz-plane. Identify the surface.

b)Find a vector equation and parametric equations for the line segment that joins P(a; b; c) to Q(u; v; w).

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(a). Let P(x,y,z) be the point in XYZ-plane such that the distance from P to the y axis is twice the distance from P to the xz-plane i.e.

If `d_1=` distance from from P(x,y,z) to y-axis and

`d_2=` distance from P(x,y,z) to xz-plane .

By given condition

`d_1=2d_2` (i)

But

`d_1=sqrt(x^2+z^2)`

`d_2=y`

Substitute `d_1 and d_2` in (i), we have

`sqrt(x^2+z^2)=2y`

`x^2+y^2-4y^2=0`

(b). Equation of the line in vectors form joining two vectors `vecX_0 and vecX_1` is

`vecX=(1-lambda)vecX_0+lambdavecX_1 , lambda in[0,1]`

`vecX=(1-lambda)(a,b,c)+lambda(u,v,w)`

`vecX=((1-lambda)a+lambdau,(1-lambda)b+lambdav,(1-lambda)c+lambdaw)`

`=(a+lambda(u-a),b+lambda(v-b),c+lambda(w-c))`

Parametric equations are

x=a+lambda(u-a)

y=b+lambda(v-b)

z=c+lambda(w-c) , lambda in [0,1].

x_(mn)

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