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

Options

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

Answer

To find the length of the perpendicular drawn from the origin (0,0) to the line given by the equation \(\frac{x}{a} + \frac{y}{b} = 1\), we can follow these steps:

### Step 1: Rewrite the line in standard form
The given line equation is:
\[ \frac{x}{a} + \frac{y}{b} = 1 \]
To convert this into the standard form \(Ax + By + C = 0\), we can rearrange it:
\[ \frac{x}{a} + \frac{y}{b} - 1 = 0 \]
Multiplying through by \(ab\) to eliminate the fractions gives:
\[ bx + ay - ab = 0 \]
Thus, we have:
\[ A = b, \quad B = a, \quad C = -ab \]

### Step 2: Use the formula for the length of the perpendicular
The formula for the length of the perpendicular \(d\) from a point \((x_1, y_1)\) to the line \(Ax + By + C = 0\) is given by:
\[ d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}} \]
In our case, the point is the origin \((0, 0)\), so \(x_1 = 0\) and \(y_1 = 0\). Plugging in these values:
\[ d = \frac{|b(0) + a(0) - ab|}{\sqrt{b^2 + a^2}} = \frac{|-ab|}{\sqrt{b^2 + a^2}} = \frac{ab}{\sqrt{b^2 + a^2}} \]

### Step 3: Conclusion
Thus, the length of the perpendicular \(p\) from the origin to the line \(\frac{x}{a} + \frac{y}{b} = 1\) is:
\[ p = \frac{ab}{\sqrt{a^2 + b^2}} \]