Question

By shifting the origin to (-1, 2) the equation \[{y^2} + 8x - 4y + 12 = 0 \] changes as \[{Y^2} = 4aX \] then \[a \] =
A.1
B.2
C.-2
D.-1

Answer 1

Hint: If (x, y) are the coordinate of a point, and the origin is shifted to (h, k), without changing the orientation of the axes, then the new coordinates (X, Y) of the point will be given by x=X+ h and y=Y+ k.
In this question two equations of the lines have been given and it is said that the origin has been shifted to (-1, 2), so we will write the new equation of the line after shifting the origin and then we will compare it by given equations to find the value of a.

Complete step-by-step answer:
Given the equation \[{y^2} + 8x - 4y + 12 = 0 - - (i) \]
Now let (x, y) be the coordinate of any point on the given line changes to (X, Y) on shifting the origin to (-1, 2)
Hence we can write the new coordinates as
x=X-1
y=Y+2
Hence we can write the given equation in the transformed as
 \[{ \left( {Y + 2} \right)^2} + 8 \left( {X - 1} \right) - 4 \left( {Y + 2} \right) + 12 = 0 - - (ii) \]
Further solving this equation we can write
 \[
 \Rightarrow {Y^2} + 4Y + 4 + 8X - 8 - 4Y - 8 + 12 = 0 \ \
 \Rightarrow {Y^2} + 8X - 16 + 16 = 0 \ \
 \Rightarrow {Y^2} = - 8X \;
  \]
Now we compare this equation with the given transformed equation \[{Y^2} = 4aX \] by the obtained transformed equation \[{Y^2} = - 8X \] , hence we can write
 \[
 \Rightarrow 4a = - 8 \ \
 \Rightarrow a = - 2 \;
  \]
Therefore the value of a after comparing the both transformed equation is \[a = - 2 \]
So, the correct answer is “a=-2”.

Note: To shift the origin vertically upward we add a constant value to the y-axis in coordinate of the initial coordinates and to shift it downward we subtract a constant value from the y-axis in coordinate of the initial coordinates.