In a `Delta ABC` right angle at C .P is the length of `_|_` from C to AB. If a, b, c, are the length of BC , CA, AB. Which relation is correct .

Options

A. `(1)/(P^(2))=(1)/(a^(2))-(1)/(b^(2))`
B. `(1)/(P^(2))=(1)/(b^(2))-(1)/(a^(2))`
C. `(1)/(P^(2))=(1)/(a^(2))+(1)/(b^(2))`
D. `(1)/(P^(2))+(1)/(a^(2))=(1)/(b^(2))=0`

Answer

To solve the problem step by step, we will analyze the right triangle ABC, where angle C is the right angle. We will derive the relationship between the sides of the triangle and the height from point C to side AB.

### Step 1: Understand the Triangle
We have a right triangle ABC with:
- Angle C = 90 degrees
- Sides:
- BC = a
- CA = b
- AB = c

### Step 2: Area of Triangle
The area of triangle ABC can be calculated using the formula:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]
We can consider different bases and corresponding heights.

### Step 3: Using Base AB
If we take AB as the base, the height from C to AB is P. Thus, the area can be expressed as:
\[ \text{Area} = \frac{1}{2} \times c \times P \]

### Step 4: Using Base BC
Now, if we take BC as the base, the height from A to BC is the same height P. Thus, the area can also be expressed as:
\[ \text{Area} = \frac{1}{2} \times a \times P \]

### Step 5: Equating the Areas
Since both expressions represent the area of the same triangle, we can equate them:
\[ \frac{1}{2} \times c \times P = \frac{1}{2} \times a \times P \]
This simplifies to:
\[ c \times P = a \times P \]
Assuming P is not zero, we can divide both sides by P:
\[ c = a \]
This is not correct since we know that c is the hypotenuse.

### Step 6: Using Pythagorean Theorem
Using the Pythagorean theorem, we know:
\[ a^2 + b^2 = c^2 \]

### Step 7: Expressing Area in Terms of Sides
We can also express the area in terms of the sides:
\[ \text{Area} = \frac{1}{2} \times a \times b \]

### Step 8: Relating Area with Height
We can also express the area using the height:
\[ \text{Area} = \frac{1}{2} \times c \times P \]
Equating the two area expressions gives:
\[ \frac{1}{2} \times a \times b = \frac{1}{2} \times c \times P \]
Cancelling \(\frac{1}{2}\) from both sides, we get:
\[ a \times b = c \times P \]

### Step 9: Deriving the Final Relation
Rearranging gives us:
\[ P = \frac{a \times b}{c} \]
Now, we can use this to derive the relationship:
Using the area again, we can express:
\[ \frac{1}{P} = \frac{1}{a} + \frac{1}{b} \]
This leads us to:
\[ \frac{1}{c^2} = \frac{1}{a^2} + \frac{1}{b^2} \]

### Final Relation
Thus, the correct relationship is:
\[ \frac{1}{c^2} = \frac{1}{a^2} + \frac{1}{b^2} \]