Assignment
Calculate the distance betwren the poind
A(2,3) and the inters and

Ask by Cox Howell. in Nigeria

Jan 26,2025

Question

Assignment Calculate the distance betwren the poind A(2,3) and the inters $$ y=x+s $$ and $$ 2 x+y=0 $$

UpStudy AI · Accepted Answer

To calculate the distance between the point $$ A(2, 3) $$ and the intersection of the two lines:

1. $$ y = x + s $$
2. $$ 2x + y = 0 $$

where $$ s $$ is a constant, follow these steps:

### 1. Find the Intersection Point of the Two Lines

To find the intersection point $$ (x, y) $$ of the lines $$ y = x + s $$ and $$ 2x + y = 0 $$, substitute $$ y = x + s $$ into the second equation:

$$
2x + (x + s) = 0
$$
$$
3x + s = 0
$$
$$
x = -\frac{s}{3}
$$

Now, substitute $$ x = -\frac{s}{3} $$ back into $$ y = x + s $$ to find $$ y $$:

$$
y = -\frac{s}{3} + s = \frac{2s}{3}
$$

So, the intersection point is:
$$
\left( -\frac{s}{3}, \frac{2s}{3} \right)
$$

### 2. Calculate the Distance Between $$ A(2, 3) $$ and the Intersection Point

The distance $$ D $$ between two points $$ (x_1, y_1) $$ and $$ (x_2, y_2) $$ is given by the formula:

$$
D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
$$

Plugging in the coordinates of $$ A(2, 3) $$ and the intersection point $$ \left( -\frac{s}{3}, \frac{2s}{3} \right) $$:

$$
D = \sqrt{\left( -\frac{s}{3} - 2 \right)^2 + \left( \frac{2s}{3} - 3 \right)^2}
$$

Simplify the expressions inside the square root:

$$
D = \sqrt{\left( -\frac{s + 6}{3} \right)^2 + \left( \frac{2s - 9}{3} \right)^2}
$$
$$
D = \sqrt{\frac{(s + 6)^2 + (2s - 9)^2}{9}}
$$
$$
D = \frac{1}{3} \sqrt{(s + 6)^2 + (2s - 9)^2}
$$

Expand the squared terms:

$$
(s + 6)^2 = s^2 + 12s + 36
$$
$$
(2s - 9)^2 = 4s^2 - 36s + 81
$$

Add them together:

$$
s^2 + 12s + 36 + 4s^2 - 36s + 81 = 5s^2 - 24s + 117
$$

Now, the distance becomes:

$$
D = \frac{1}{3} \sqrt{5s^2 - 24s + 117}
$$

### **Final Formula for the Distance**

$$
D = \frac{\sqrt{5s^2 - 24s + 117}}{3}
$$

### **Example Calculation**

If a specific value for $$ s $$ is given, you can substitute it into the formula to find the numerical distance. For instance, let's assume $$ s = 3 $$:

$$
D = \frac{\sqrt{5(3)^2 - 24(3) + 117}}{3} = \frac{\sqrt{45 - 72 + 117}}{3} = \frac{\sqrt{90}}{3} = \frac{3\sqrt{10}}{3} = \sqrt{10} \approx 3.162
$$

So, when $$ s = 3 $$, the distance $$ D $$ is approximately $$ 3.162 $$ units.

---

**Note:** If $$ s $$ is a known constant in your assignment, replace it with its value to compute the specific distance. If $$ s $$ is meant to represent something else (like a variable parameter), ensure you understand its intended use within the problem context.
The distance between point $$ A(2, 3) $$ and the intersection of the lines $$ y = x + s $$ and $$ 2x + y = 0 $$ is $$ \frac{\sqrt{5s^2 - 24s + 117}}{3} $$.

Assignment
Calculate the distance betwren the poind
A(2,3) and the inters and

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Assignment Calculate the distance betwren the poind A(2,3) and the inters and