Consider the following statements :
1 . For an equation of a line x cos q + y sin q = p , in normal form the length of the perpendicular from the point (a , b) to the line is |a cos q + b sin q + p|
2 The length of the perpendicular from the point `(alpha , beta)` to the line `(x)/(a) + (y)/(b) = 1` is `|(a alpha + b beta -ab)/(sqrt(a^(2) + b^(2)))|`
Which of the above statements is/are correct ?

Options

A. 1 only
B. 2 only
C. Both 1 and 2
D. Neither 1 nor 2

Answer

To determine the correctness of the given statements about the lengths of perpendiculars from points to lines, we will analyze each statement step by step.

### Step 1: Analyze Statement 1
The first statement claims that for the line given in normal form \( x \cos q + y \sin q = p \), the length of the perpendicular from the point \( (a, b) \) to this line is \( |a \cos q + b \sin q + p| \).

**Solution:**
1. The equation of the line can be rewritten as:
\[ x \cos q + y \sin q - p = 0 \]
2. The formula for the length of the perpendicular from a point \( (x_0, y_0) \) to the line \( Ax + By + C = 0 \) is given by:
\[ \text{Length} = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}} \]
3. Here, \( A = \cos q \), \( B = \sin q \), and \( C = -p \). Substituting \( (x_0, y_0) = (a, b) \):
\[ \text{Length} = \frac{|a \cos q + b \sin q - p|}{\sqrt{\cos^2 q + \sin^2 q}} = |a \cos q + b \sin q - p| \]
4. The statement claims \( |a \cos q + b \sin q + p| \), which is incorrect. The correct expression should have a negative \( p \).

**Conclusion for Statement 1:** Incorrect.

### Step 2: Analyze Statement 2
The second statement claims that the length of the perpendicular from the point \( (\alpha, \beta) \) to the line \( \frac{x}{a} + \frac{y}{b} = 1 \) is given by \( \left| \frac{a \alpha + b \beta - ab}{\sqrt{a^2 + b^2}} \right| \).

**Solution:**
1. Rewrite the line equation:
\[ \frac{x}{a} + \frac{y}{b} - 1 = 0 \implies bx + ay - ab = 0 \]
2. Here, \( A = b \), \( B = a \), and \( C = -ab \).
3. Using the formula for the length of the perpendicular:
\[ \text{Length} = \frac{|b\alpha + a\beta - ab|}{\sqrt{b^2 + a^2}} \]
4. The statement claims \( |a\alpha + b\beta - ab| \), which is incorrect. The correct expression should have \( b\alpha + a\beta \).

**Conclusion for Statement 2:** Incorrect.

### Final Conclusion
Both statements are incorrect.