4. Find the coordinates of the point on the linie presented by $$ 2 x-3 y+z=0 $$ which is equidistant from the point $$ (-4,-8) $$ and $$ (7,-1) $$

Question

UpStudy AI · Accepted Answer

To find the coordinates of the point on the line $$ 2x - 3y = 0 $$ that is equidistant from the points $$ A(-4, -8) $$ and $$ B(7, -1) $$, follow these steps:

### 1. **Understand the Problem**

We need to determine a point $$ P(x, y) $$ on the line $$ 2x - 3y = 0 $$ such that the distance from $$ P $$ to $$ A $$ is equal to the distance from $$ P $$ to $$ B $$.

### 2. **Set Up the Distance Equation**

The distance from $$ P(x, y) $$ to $$ A(-4, -8) $$ is:

$$
\sqrt{(x + 4)^2 + (y + 8)^2}
$$

The distance from $$ P(x, y) $$ to $$ B(7, -1) $$ is:

$$
\sqrt{(x - 7)^2 + (y + 1)^2}
$$

Setting these two distances equal:

$$
\sqrt{(x + 4)^2 + (y + 8)^2} = \sqrt{(x - 7)^2 + (y + 1)^2}
$$

### 3. **Square Both Sides to Eliminate the Square Roots**

$$
(x + 4)^2 + (y + 8)^2 = (x - 7)^2 + (y + 1)^2
$$

Expanding both sides:

$$
x^2 + 8x + 16 + y^2 + 16y + 64 = x^2 - 14x + 49 + y^2 + 2y + 1
$$

Simplifying:

$$
8x + 16y + 80 = -14x + 2y + 50
$$

Rearranging terms:

$$
22x + 14y = -30 \quad \text{(Equation 1)}
$$

### 4. **Combine with the Line Equation**

We also have the line equation:

$$
2x - 3y = 0 \quad \text{(Equation 2)}
$$

### 5. **Solve the System of Equations**

From Equation 2:

$$
2x = 3y \implies x = \frac{3y}{2}
$$

Substitute $$ x = \frac{3y}{2} $$ into Equation 1:

$$
22\left(\frac{3y}{2}\right) + 14y = -30
$$

$$
33y + 14y = -30
$$

$$
47y = -30 \implies y = -\frac{30}{47}
$$

Now, find $$ x $$:

$$
x = \frac{3(-30/47)}{2} = -\frac{45}{47}
$$

### 6. **Solution**

The coordinates of the required point are:

$$
\boxed{\left( -\dfrac{45}{47},\; -\dfrac{30}{47} \right)}
$$
The point on the line $$ 2x - 3y = 0 $$ that is equidistant from $$ (-4, -8) $$ and $$ (7, -1) $$ is $$ \left( -\dfrac{45}{47},\; -\dfrac{30}{47} \right) $$.

Find the coordinates of the
point on the linie presented
by which is
equidistant from the point
and

Upstudy AI Solution

Answer

Solution

1. Understand the Problem

2. Set Up the Distance Equation

3. Square Both Sides to Eliminate the Square Roots

4. Combine with the Line Equation

5. Solve the System of Equations

6. Solution

Bonus Knowledge

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