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.