Find the value of p, if the line 3x + 4y – p = 0 is a tangent to the circle 2 2 16 + = x y . 

To solve for \( p \), we are given that the line \( 3x + 4y - p = 0 \) is a tangent to the circle \( x^2 + y^2 = 16 \).

### Step 1: Equation of the circle
The equation of the circle is:
\[
x^2 + y^2 = 16
\]
This is a circle with center at \( (0, 0) \) and radius \( r = \sqrt{16} = 4 \).

### Step 2: Equation of the line
The equation of the line is:
\[
3x + 4y - p = 0
\]
We can rewrite this as:
\[
3x + 4y = p
\]
This line is in the form \( Ax + By + C = 0 \), where \( A = 3 \), \( B = 4 \), and \( C = -p \).

### Step 3: Condition for tangency
For a line to be tangent to a circle, the perpendicular distance from the center of the circle to the line must be equal to the radius of the circle.

The formula for the perpendicular distance \( d \) 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 this case, the center of the circle is \( (0, 0) \), so the distance from the center to the line is:
\[
d = \frac{|3(0) + 4(0) - p|}{\sqrt{3^2 + 4^2}} = \frac{| - p |}{\sqrt{9 + 16}} = \frac{|p|}{5}
\]
Since the line is a tangent, this distance must be equal to the radius of the circle, which is 4. Thus:
\[
\frac{|p|}{5} = 4
\]
Solving for \( p \):
\[
|p| = 20
\]

### Step 4: Final answer
Since \( |p| = 20 \), \( p \) can be either \( 20 \) or \( -20 \).

Thus, the value of \( p \) is \( \boxed{20} \) or \( \boxed{-20} \).

Would you like more details or have any questions?

Here are 5 related questions:
1. What is the general formula for the perpendicular distance between a point and a line?
2. How do you find the equation of a tangent line to a circle at a given point?
3. What is the geometric interpretation of a line being tangent to a circle?
4. How do you derive the equation of a circle given its center and radius?
5. What are the conditions for two lines to be perpendicular or parallel?

**Tip:** When working with tangents, always check that the perpendicular distance from the center of the circle to the line equals the radius.

Find the value of p, if the line 3x + 4y – p = 0 is a tangent to the circle x^2 + y^2 = 16.

Learn how to find the value of p for the line 3x + 4y - p = 0, which is a tangent to the circle x^2 + y^2 = 16. This problem involves key geometric concepts such as tangents, circles, and the formula for perpendicular distance. Suitable for high school students in Grades 9-12.

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