https://file.aiquickdraw.com/mathbot/file/17274495790181bl4g5y1.jpg

The question in the image is in Gujarati, and it reads:

**(-5, -2) અને (3, 2) થી સમાન અંતરે હોય તેવું y-અક્ષ પરનું વિંદુ શોધો.**

Translation: 

"Find the point on the y-axis that is equidistant from the points (-5, -2) and (3, 2)."

### Solution:

To find a point on the y-axis that is equidistant from two given points, let the coordinates of the point on the y-axis be \( (0, y) \).

We need to find \( y \) such that the distance from \( (0, y) \) to both \( (-5, -2) \) and \( (3, 2) \) is the same.

#### Step 1: Distance Formula

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

\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]

#### Step 2: Set the two distances equal

1. Distance between \( (0, y) \) and \( (-5, -2) \):
\[
d_1 = \sqrt{(0 - (-5))^2 + (y - (-2))^2} = \sqrt{5^2 + (y + 2)^2} = \sqrt{25 + (y + 2)^2}
\]

2. Distance between \( (0, y) \) and \( (3, 2) \):
\[
d_2 = \sqrt{(0 - 3)^2 + (y - 2)^2} = \sqrt{3^2 + (y - 2)^2} = \sqrt{9 + (y - 2)^2}
\]

#### Step 3: Equate the two distances

\[
\sqrt{25 + (y + 2)^2} = \sqrt{9 + (y - 2)^2}
\]

Squaring both sides:

\[
25 + (y + 2)^2 = 9 + (y - 2)^2
\]

Expanding both sides:

\[
25 + y^2 + 4y + 4 = 9 + y^2 - 4y + 4
\]

Simplify:

\[
29 + y^2 + 4y = 13 + y^2 - 4y
\]

Cancel out \( y^2 \) on both sides:

\[
29 + 4y = 13 - 4y
\]

Add \( 4y \) to both sides:

\[
29 + 8y = 13
\]

Subtract 29 from both sides:

\[
8y = 13 - 29
\]

\[
8y = -16
\]

Solve for \( y \):

\[
y = \frac{-16}{8} = -2
\]

Thus, the point on the y-axis is \( (0, -2) \).

### Final Answer:
The point on the y-axis equidistant from \( (-5, -2) \) and \( (3, 2) \) is \( (0, -2) \).

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Would you like more details or have any questions? Here are some related questions:

1. How do we calculate the distance between two points in 2D space?
2. What other methods can we use to find equidistant points on different axes?
3. How does the y-axis simplify finding the equidistant point?
4. Can we use symmetry to predict the point on the y-axis?
5. How would this change if the point needed to be on the x-axis instead?

**Tip:** When finding points equidistant from two points, always use the distance formula and look for symmetry in the problem to simplify calculations.

Find the point on the y-axis that is equidistant from the points (-5, -2) and (3, 2).

This problem helps students learn how to find a point on the y-axis that is equidistant from two given points, using the distance formula. Suitable for Grades 9-12, it covers key concepts in coordinate geometry and equidistance.

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