Line 3x + 4y = 25 touches the circle `x^(2) + y^(2) = 25` at the point

Options

A. (4, 3)
B. (3, 4)
C. (-3, -4)
D. none of these

Answer

To find the point of contact where the line \(3x + 4y = 25\) touches the circle \(x^2 + y^2 = 25\), we can follow these steps:

### Step 1: Identify the equations
We have the equation of the line:
\[ 3x + 4y = 25 \]
And the equation of the circle:
\[ x^2 + y^2 = 25 \]

### Step 2: Rewrite the line equation in slope-intercept form
To find the slope of the line, we can rewrite the line equation in the form \(y = mx + b\):
\[ 4y = -3x + 25 \implies y = -\frac{3}{4}x + \frac{25}{4} \]
The slope of the line is \(-\frac{3}{4}\).

### Step 3: Find the radius and center of the circle
The circle \(x^2 + y^2 = 25\) has its center at the origin \((0, 0)\) and a radius \(r = 5\) (since \(\sqrt{25} = 5\)).

### Step 4: Use the point of tangency condition
Let the point of tangency be \((x_1, y_1)\). The distance from the center of the circle to the line must equal the radius. The distance \(d\) from a point \((x_0, y_0)\) to the line \(Ax + By + C = 0\) is given by:
\[ d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}} \]
For our line \(3x + 4y - 25 = 0\), we have \(A = 3\), \(B = 4\), and \(C = -25\). The center of the circle is \((0, 0)\).

### Step 5: Calculate the distance from the center to the line
Substituting into the distance formula:
\[ d = \frac{|3(0) + 4(0) - 25|}{\sqrt{3^2 + 4^2}} = \frac{|-25|}{\sqrt{9 + 16}} = \frac{25}{5} = 5 \]
Since the distance \(d\) is equal to the radius \(r = 5\), the line is indeed tangent to the circle.

### Step 6: Find the coordinates of the point of tangency
Using the point of tangency condition, we can express the coordinates of the point of tangency \((x_1, y_1)\) in terms of a parameter \(k\):
\[ x_1 = 5 \cos \theta, \quad y_1 = 5 \sin \theta \]
Substituting these into the line equation:
\[ 3(5 \cos \theta) + 4(5 \sin \theta) = 25 \]
This simplifies to:
\[ 15 \cos \theta + 20 \sin \theta = 25 \]
Dividing through by 5:
\[ 3 \cos \theta + 4 \sin \theta = 5 \]

### Step 7: Solve for \(\theta\)
This equation can be solved using the method of substitution or by recognizing that it represents a linear combination of sine and cosine. The maximum value of \(3 \cos \theta + 4 \sin \theta\) is \(\sqrt{3^2 + 4^2} = 5\), which occurs when:
\[ \tan \theta = \frac{4}{3} \]
Thus, we can find:
\[ \cos \theta = \frac{3}{5}, \quad \sin \theta = \frac{4}{5} \]

### Step 8: Calculate \(x_1\) and \(y_1\)
Substituting back:
\[ x_1 = 5 \cdot \frac{3}{5} = 3, \quad y_1 = 5 \cdot \frac{4}{5} = 4 \]

### Conclusion
Thus, the point of contact where the line touches the circle is:
\[ \boxed{(3, 4)} \]