Equation of line $2x-3y+5=0$ is change into which new line, if origin is shifted to (3,-1)?

Question

Accepted Answer

To determine the new equation of the line when the origin is shifted to (3, -1), we need to apply the concept of translation of axes. The original equation of the line is given by:

$$ 2x - 3y + 5 = 0. $$

When the origin is shifted to a new point (h, k) = (3, -1), the transformation of coordinates is given by:

$$ x' = x - h = x - 3, $$
$$ y' = y - k = y + 1. $$

We substitute these transformations into the original equation:

$$ 2(x' + 3) - 3(y' - 1) + 5 = 0. $$

Expanding this, we get:

$$ 2x' + 6 - 3y' + 3 + 5 = 0. $$

Simplifying further:

$$ 2x' - 3y' + 14 = 0. $$

Thus, the new equation of the line after the origin is shifted to (3, -1) is:

$$ 2x' - 3y' + 14 = 0. $$

This corresponds to Option 2: $ 2x - 3y + 14 = 0 $.

**Explanation for why each option is incorrect:**

1. **Option 1: $ 2x + 3y + 5 = 0 $**

This option does not account for the correct transformation of the y-term and the constant term after the shift of origin. The signs and values do not match the derived equation.

2. **Option 3: $ -2x - 3y + 14 = 0 $**

This option incorrectly changes the sign of the x-term and y-term, which is not consistent with the transformation process. The derived equation maintains the sign of the x-term and y-term as in the original equation.

3. **Option 4: $ x + y = 0 $**

This option is a completely different equation and does not relate to the transformation of the original line equation. It does not reflect the changes due to the shift of origin.

Therefore, Option 2 is the correct answer as it accurately represents the new equation of the line after the origin is shifted to (3, -1).