Question 1:What does the equation become when the axes are transferred to parallel axes through the point ?

Answer:

Substituting in the equation we get

Hence the equation gets transformed to

Question 2: What does the equation become if the origin is shifted to the point without rotation?

Answer:

Substituting in the equation we get

Hence the transformed equation is

Question 3: Find what the following equations become when the origin is shifted to the point ?

i)

ii)

iii)

iv)

Answer:

i) Substituting in the equation we get

Hence the transformed equation is

ii) Substituting in the equation we get

Hence the transformed equation is

iii) Substituting in the equation we get

Hence the transformed equation is

iv) Substituting in the equation we get

Hence the transformed equation is

Question 4: At what point the origin be shifted so that the equation does not contain any first degree term and constant term?

Answer:

Let the origin be shifted to .

Therefore

Substituting in the equation we get

For the equation to be free of 1st degree term and constant term we get

and

Also and satisfies .

Therefore origin should be shifted to .

Question 5: Verify that the area of the triangle with vertices and remains invariant under the translation of axes when the origin is shifted to the point .

Answer:

Given with vertices and

Area of a

Now we shift the origin to

Therefore the new vertices or

Therefore

Area of a

Hence the area of the triangle would remain invariant.

Question 6: Find, what the following equations become when the origin is shifted to the point .

i)

ii)

iii)

iv)

Answer:

i) Substitute in the equation we get

Hence the equation become when origin is shifted to

ii) Substitute in the equation we get

Hence the equation become when origin is shifted to

iii) Substitute in the equation we get

Hence the equation become when origin is shifted to

iv) Substitute in the equation we get

Hence the equation become when origin is shifted to

Question 7: Find the point to which the origin should be shifted after a translation of axes so that the following equations will have no first degree terms:

i)

ii)

iii)

Answer:

i) Let the origin be shifted to .

Therefore

Substituting in the equation we get

For this equation to be free of terms containing and we must have

and

and

Hence the origin should be shifted to

ii) Let the origin be shifted to .

Therefore

Substituting in the equation we get

For this equation to be free of terms containing and we must have

and

and

Hence the origin should be shifted to

iii) Let the origin be shifted to .

Therefore

Substituting in the equation we get

For this equation to be free of terms containing and we must have

Hence the origin should be shifted to

Question 8: Verify that the area of the triangle with vertices and remains invariant under the translation of axes when the origin is shifted to the point .

Answer:

Given with vertices and

Area of a

Now we shift the origin to

Therefore the new vertices or

Therefore

Area of a

Hence the area of the triangle would remain invariant.