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 $(0, -2)$.

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.