The line segment joining `A(6, 3)` to `B(-1, -4)` is doubled in length by having its length added to each end , then the ordinates of new ends are

Answer

To solve the problem of finding the ordinates of the new ends of the line segment joining points A(6, 3) and B(-1, -4) after doubling its length, we can follow these steps:

### Step 1: Calculate the Length of the Original Line Segment
The length of the line segment AB can be calculated using the distance formula:
\[ \text{Length} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Here, \(A(6, 3)\) and \(B(-1, -4)\):
\[ \text{Length} = \sqrt{((-1) - 6)^2 + ((-4) - 3)^2} = \sqrt{(-7)^2 + (-7)^2} = \sqrt{49 + 49} = \sqrt{98} = 7\sqrt{2} \]

### Step 2: Determine the New Length After Doubling
Since we need to double the length of the segment, we multiply the original length by 2:
\[ \text{New Length} = 2 \times 7\sqrt{2} = 14\sqrt{2} \]

### Step 3: Find the Midpoint of the Original Segment
The midpoint \(M\) of segment AB can be found using the midpoint formula:
\[ M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) \]
Substituting the coordinates of A and B:
\[ M = \left(\frac{6 + (-1)}{2}, \frac{3 + (-4)}{2}\right) = \left(\frac{5}{2}, \frac{-1}{2}\right) \]

### Step 4: Calculate the Direction Vector from A to B
The direction vector \( \vec{d} \) from A to B is given by:
\[ \vec{d} = (x_2 - x_1, y_2 - y_1) = (-1 - 6, -4 - 3) = (-7, -7) \]

### Step 5: Normalize the Direction Vector
To normalize the direction vector, we divide by its length:
\[ \text{Length of } \vec{d} = \sqrt{(-7)^2 + (-7)^2} = 7\sqrt{2} \]
The unit vector \( \hat{d} \) is:
\[ \hat{d} = \left(-1, -1\right) \text{ (normalized direction vector)} \]

### Step 6: Calculate the New Endpoints
To find the new endpoints, we need to move from the midpoint \(M\) in both directions along the direction vector scaled by half the new length:
\[ \text{New Length from Midpoint} = \frac{14\sqrt{2}}{2} = 7\sqrt{2} \]
The new coordinates of the endpoints can be calculated as:
1. New point \(C\):
\[ C = M + 7\sqrt{2} \cdot \hat{d} = \left(\frac{5}{2} - 7, \frac{-1}{2} - 7\right) = \left(\frac{5}{2} - \frac{14}{2}, \frac{-1}{2} - \frac{14}{2}\right) = \left(-\frac{9}{2}, -\frac{15}{2}\right) \]
2. New point \(D\):
\[ D = M - 7\sqrt{2} \cdot \hat{d} = \left(\frac{5}{2} + 7, \frac{-1}{2} + 7\right) = \left(\frac{5}{2} + \frac{14}{2}, \frac{-1}{2} + \frac{14}{2}\right) = \left(\frac{19}{2}, \frac{13}{2}\right) \]

### Step 7: Identify the Ordinates
The ordinates of the new endpoints \(C\) and \(D\) are:
- For point \(C\): \(y = -\frac{15}{2}\)
- For point \(D\): \(y = \frac{13}{2}\)

### Final Answer
The ordinates of the new ends are:
\[ \frac{13}{2} \text{ and } -\frac{15}{2} \]