STATEMENT-1 : The orthogonal trajectory of a family of circles touching x-axis at origin and whose centre the on y-axis is self orthogonal.
and
STATEMENT-2 : In order to find the orthogonal trajectory of a family of curves we put `-(dx)/(dy)` in place of `(dy)/(dx)` in the differential equation of the given family of curves.

Options

A. (a)Statement-1 is true, Statement-2 is true, Statement-2 is a correct explanation for Statement-1.
B. (b)Statement-1 is true, Statement-2 is true, Statement-2 is NOT a correct explanation for Statement-1.
C. (c)Statement-1 is True, Statement-2 is False
D. (d)Statement-1 is False, Statement-2 is True

Answer

To solve the problem, we need to analyze the two statements provided and determine their validity.

### Step 1: Understanding Statement 1
**Statement 1:** The orthogonal trajectory of a family of circles touching the x-axis at the origin and whose centers are on the y-axis is self-orthogonal.

1. **Family of Circles:** The circles that touch the x-axis at the origin and have their centers on the y-axis can be represented by the equation:
\[ x^2 + (y - k)^2 = k^2 \]
where \(k\) is the y-coordinate of the center of the circle.

2. **Condition for Touching the x-axis:** For the circle to touch the x-axis at the origin, the radius \(r = k\). Therefore, the equation simplifies to:
\[ x^2 + (y - k)^2 = k^2 \]

3. **Finding the Orthogonal Trajectory:** To find the orthogonal trajectory, we differentiate the equation of the circle with respect to \(x\) and replace \(\frac{dy}{dx}\) with \(-\frac{dx}{dy}\).

4. **Differentiating the Circle Equation:**
\[ 2x + 2(y - k)\frac{dy}{dx} = 0 \]
Rearranging gives:
\[ \frac{dy}{dx} = -\frac{x}{y - k} \]

5. **Replacing \(\frac{dy}{dx}\):**
\[ -\frac{dx}{dy} = -\frac{x}{y - k} \implies \frac{dx}{dy} = \frac{x}{y - k} \]

6. **Solving the Differential Equation:** This leads to a separable differential equation. Solving it will show whether the resulting family of curves is self-orthogonal.

### Step 2: Understanding Statement 2
**Statement 2:** In order to find the orthogonal trajectory of a family of curves, we put \(-\frac{dx}{dy}\) in place of \(\frac{dy}{dx}\) in the differential equation of the given family of curves.

1. **Correctness of Statement 2:** This statement is true. The orthogonal trajectories are found by replacing \(\frac{dy}{dx}\) with \(-\frac{dx}{dy}\) in the original differential equation.

### Conclusion:
- **Statement 1** is **false** because the orthogonal trajectory derived from the family of circles does not yield a self-orthogonal family.
- **Statement 2** is **true** as it correctly describes the method to find orthogonal trajectories.

Thus, the correct option is:
- **Option D:** Statement 1 is false, Statement 2 is true.