Question

Find the equation of the curve $2{x^2} + {y^2} - 3x + 5y - 8 = 0$ when origin is transferred to the point (-1, 2) without changing the direction of the axis.

Hint: To solve the above problem we need to know the concept of transformation of the axis, because the question says we have to find the equation of the curve when it has been transferred from the origin to the given point which is the transformation concept.

Let us consider the given as old equation
Old equation is $\Rightarrow$ $2{x^2} + {y^2} - 3x + 5y - 8 = 0 - - - - - - - > (1)$
From the old equation we say the (x, y) are the old coordinates
Now let (X, Y) be the new coordinates after shifting from origin to new point P (-1, 2) let the point be (h, k).
Now by the concept of transformation of axis we know that
$x = X + h$ $y = Y + K$
Now from using the above concept we can say that,
$x = X - 1$ $y = Y + 2$$- - - - - - - - - >$(New values)
Now on substituting the new values in equation we get we can write the equation as
$\Rightarrow 2{(X - 1)^2} + {(Y + 2)^2} - 3(X - 1) + 5(Y + 2) = 0 \\ \Rightarrow 2{X^2} + 1 - 4X + {Y^2} + 4 + 4Y - 3X + 3 + 5Y + 10 = 0 \\ \Rightarrow 2{X^2} + {Y^2} - 7X + 9Y + 18 = 0 \\$
Thus they mentioned that direction of axis has not changed we have to replace the (X,Y) with (x ,y)
On replacing the values we get equation in form $2{x^2} + {y^2} - 7x + 9y + 18 = 0$
Hence the equation is of ellipse.

Note: In this problem we have considered the coordinates of given equation as old coordinates .And they mentioned that the curve of the equation has been shifted from origin to point p which we have considered as (h, k).Now by using the concept of transformation we have solved the equation. Finally it is mentioned that transformation has been done without changing the direction so as we have replaced the new coordinates with old coordinates.