If p is the length of the perpendicular from the origin to the line `(x)/(a) + (y)/(b) = 1`, then which of the following is ture ?

Options

A. `(1)/(p^(2)) = (1)/(b^(2)) - (1)/(a^(2))`
B. `(1)/(p^(2))=(1)/(a^(2))-(1)/(b^(2))`
C. `(1)/(p^(2))=(1)/(a^(2))+(1)/(b^(2))`
D. None of these

Answer

To find the length of the perpendicular \( p \) from the origin to the line given by the equation

\[ \frac{x}{a} + \frac{y}{b} = 1, \]

we will follow these steps:

### Step 1: Rewrite the equation in standard form
We start by rewriting the given equation in the standard form \( Ax + By + C = 0 \).

\[ \frac{x}{a} + \frac{y}{b} = 1 \implies bx + ay - ab = 0. \]

Here, we can identify \( A = b \), \( B = a \), and \( C = -ab \).

### Step 2: Use the formula for the distance from a point to a line
The formula for the distance \( d \) from a point \( (x_0, y_0) \) to the line \( Ax + By + C = 0 \) is given by:

\[ d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}}. \]

### Step 3: Substitute the origin coordinates
Since we want the distance from the origin \( (0, 0) \), we substitute \( x_0 = 0 \) and \( y_0 = 0 \):

\[ d = \frac{|b(0) + a(0) - ab|}{\sqrt{b^2 + a^2}} = \frac{| - ab |}{\sqrt{b^2 + a^2}} = \frac{ab}{\sqrt{b^2 + a^2}}. \]

### Step 4: Relate \( p \) to the distance \( d \)
Thus, we have:

\[ p = \frac{ab}{\sqrt{b^2 + a^2}}. \]

### Step 5: Find \( \frac{1}{p^2} \)
Now, we need to find \( \frac{1}{p^2} \):

\[ p^2 = \left(\frac{ab}{\sqrt{b^2 + a^2}}\right)^2 = \frac{a^2b^2}{b^2 + a^2}. \]

Thus,

\[ \frac{1}{p^2} = \frac{b^2 + a^2}{a^2b^2}. \]

### Step 6: Simplify \( \frac{1}{p^2} \)
We can separate the terms:

\[ \frac{1}{p^2} = \frac{1}{a^2} + \frac{1}{b^2}. \]

### Conclusion
Therefore, the correct statement is:

\[ \frac{1}{p^2} = \frac{1}{a^2} + \frac{1}{b^2}. \]

This corresponds to Option 3.

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