The coordinate axes is rotated and shifted in such a way that the `"IV"^("th")` quadrant direction of line `4x+3y-35=0` becomes that new positive x - axis direction and the `"I"^("st")` quadrant direction of line `3x-4y+5=0` becomes the new positive y - axis direction. If origin as per old coordinate system is O, then according to the new coordinate system, the coordinates of O are

Options

A. `(1, 7)`
B. `(-1, 7)`
C. `(1, -7)`
D. `(7, -1)`

Answer

To solve the problem, we need to find the new coordinates of the origin (0, 0) after the coordinate axes have been rotated and shifted based on the given lines. Let's break down the solution step by step.

### Step 1: Identify the lines and their equations
The two lines given are:
1. Line 1: \(4x + 3y - 35 = 0\)
2. Line 2: \(3x - 4y + 5 = 0\)

### Step 2: Calculate the distance of the origin from Line 1
The formula for the distance \(d\) from a point \((x_0, y_0)\) to a line \(Ax + By + C = 0\) is given by:
\[ d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}} \]
For Line 1, \(A = 4\), \(B = 3\), and \(C = -35\). The coordinates of the origin are \((0, 0)\).

Substituting these values into the distance formula:
\[ d_1 = \frac{|4(0) + 3(0) - 35|}{\sqrt{4^2 + 3^2}} = \frac{|-35|}{\sqrt{16 + 9}} = \frac{35}{\sqrt{25}} = \frac{35}{5} = 7 \]

### Step 3: Calculate the distance of the origin from Line 2
For Line 2, \(A = 3\), \(B = -4\), and \(C = 5\).

Using the distance formula again:
\[ d_2 = \frac{|3(0) - 4(0) + 5|}{\sqrt{3^2 + (-4)^2}} = \frac{|5|}{\sqrt{9 + 16}} = \frac{5}{\sqrt{25}} = \frac{5}{5} = 1 \]

### Step 4: Determine the new coordinates of the origin
According to the problem, the direction of Line 1 (which is in the fourth quadrant) becomes the new positive x-axis, and the direction of Line 2 (which is in the first quadrant) becomes the new positive y-axis.

From our calculations:
- The distance from the origin to Line 1 is 7. Since this line is in the fourth quadrant and becomes the new x-axis, the x-coordinate of the new origin will be 7.
- The distance from the origin to Line 2 is 1. Since this line is in the first quadrant and becomes the new y-axis, the y-coordinate of the new origin will be -1 (as it is below the new x-axis).

Thus, the new coordinates of the origin \(O\) in the new coordinate system are:
\[ O' = (1, -7) \]

### Final Answer
The coordinates of O according to the new coordinate system are \((1, -7)\).

---