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Let v0=a,band let v=x,yDescribe the sets of points in R2 satisfying the following properties:

(a).v=4(b).v4(c).v-v0=4

Short Answer

Expert verified

(a) The set of points are (0,4),(0,-4),(4,0)and(-4,0)etc.

(b) The set of values of x,y, which satisfy the relation x2+y216will be the required set of points.

The set of points are role="math" localid="1663552322576" (0,1),(0,2),(0,3),(1,1),(1,2),(1,3),(2,1),(2,2),(3,1),(0,4),(0,-4),(2,3)etc.

(c) The set of values of (x,y)and(a,b), which satisfy the relation (x-a)2+(y-b)2=16will be the required set of points.

The set of points (x,y)are role="math" localid="1663552768845" (5,1),(4,1)and(1,5)etc.

The set of points(a,b)arerole="math" localid="1663552801181" (1,1),(0,1)etc.

Step by step solution

01

Part (a)

Given that, v=(x,y)

role="math" localid="1663552119670" v=x2+y2

role="math" localid="1663552143759" x2+y2=4x2+y2=16(Squaringbothside)

When x=0,

y=16y=±4

When y=0,

x=16x=±4

Thus, the set of points are(0,4),(0,-4),(4,0)and(-4,0)etc.

02

Part (b)

Given that, v=(x,y)

v=x2+y2

x2+y24x2+y216(Squaringbothside)

All the points that satisfy the above relationship, will be the required sets of points.

The set of points are,

(0,1),(0,2),(0,3),(1,1),(1,2),(1,3),(2,1),(2,2),(3,1),(0,4),(0,-4),(2,3)etc.

03

Part (c)

Given that,

v=(x,y)andv0=(a,b)v-v0=(x-a),(y-b)v-v0=(x-a)2+(y-b)2(x-a)2+(y-b)2=4Given(x-a)2+(y-b)2=16Squaringbothside

All the values of (x,y)and(a,b)that satisfy the above relationship will be the required sets of points.

The values of (x,y)are(5,1),(4,1)and(1,5)etc.

The values of(a,b)are(1,1),(0,1)etc.

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