Statement-I : The condition for the line y = mx+c to be a tangent to `(x+a)^(2)` = 4ay is c = am(1-m).
Statement-II : The condition for tbe line y = mx + c to be a focal chord to `y^(2) = 4ax` is c+am=0
Statement-III : The condition for the line y = mx + c to be a tangent `x^(2)=4ay` is c = -`am^(2)`
Which of above stattements is true

Options

A. only I
B. only III
C. both I, II & III
D. neither I nor II

Answer

To determine the truth of the statements regarding the conditions for a line to be tangent or a focal chord to a parabola, let's analyze each statement step by step.

### Step 1: Analyze Statement I
**Statement I:** The condition for the line \( y = mx + c \) to be a tangent to \( (x + a)^2 = 4ay \) is \( c = am(1 - m) \).

1. Start with the parabola equation:
\[ (x + a)^2 = 4ay \]
Rearranging gives:
\[ y = \frac{(x + a)^2}{4a} \]

2. Substitute \( y = mx + c \) into the parabola equation:
\[ (x + a)^2 = 4a(mx + c) \]
Expanding both sides:
\[ x^2 + 2ax + a^2 = 4amx + 4ac \]

3. Rearranging gives a quadratic equation in \( x \):
\[ x^2 + (2a - 4am)x + (a^2 - 4ac) = 0 \]

4. For the line to be tangent to the parabola, the discriminant must be zero:
\[ b^2 - 4ac = 0 \]
Here, \( b = 2a - 4am \) and \( c = a^2 - 4ac \).

5. Calculate the discriminant:
\[ (2a - 4am)^2 - 4(1)(a^2 - 4ac) = 0 \]

6. Simplifying leads to:
\[ 4a^2(1 - 2m)^2 - 4(a^2 - 4ac) = 0 \]
This simplifies to:
\[ 4a^2(1 - 2m)^2 = 4a^2 - 16ac \]

7. Solving gives:
\[ c = am(1 - m) \]
Thus, **Statement I is true.**

### Step 2: Analyze Statement II
**Statement II:** The condition for the line \( y = mx + c \) to be a focal chord to \( y^2 = 4ax \) is \( c + am = 0 \).

1. The parabola \( y^2 = 4ax \) has its focus at \( (a, 0) \).

2. For the line to be a focal chord, it must pass through the focus:
\[ 0 = ma + c \implies c = -am \]

3. Thus, rewriting gives:
\[ c + am = 0 \]
Therefore, **Statement II is true.**

### Step 3: Analyze Statement III
**Statement III:** The condition for the line \( y = mx + c \) to be tangent to \( x^2 = 4ay \) is \( c = -am^2 \).

1. Start with the parabola equation:
\[ x^2 = 4ay \]
Rearranging gives:
\[ y = \frac{x^2}{4a} \]

2. Substitute \( y = mx + c \) into the parabola:
\[ x^2 = 4a(mx + c) \]
Expanding gives:
\[ x^2 - 4amx - 4ac = 0 \]

3. The discriminant must be zero for tangency:
\[ (-4am)^2 - 4(1)(-4ac) = 0 \]

4. Simplifying gives:
\[ 16a^2m^2 + 16ac = 0 \]

5. This leads to:
\[ am^2 + c = 0 \implies c = -am^2 \]
Thus, **Statement III is true.**

### Conclusion
All statements are true:
- **Statement I:** True
- **Statement II:** True
- **Statement III:** True