Tangents are drawn to the circle `x^(2)+y^(2)=9` at the points where it is cut by the line `4x+3y-9=0` then the point of intersection of tangents is

Options

A. (3,4)
B. (4,3)
C. (-3,4)
D. (4,-3)

Answer

To solve the problem, we need to find the point of intersection of the tangents drawn to the circle \(x^2 + y^2 = 9\) at the points where it is cut by the line \(4x + 3y - 9 = 0\).

### Step-by-Step Solution:

1. **Identify the Circle and Line:**
The equation of the circle is given by:
\[ x^2 + y^2 = 9 \]
This circle has a center at the origin (0, 0) and a radius of 3 (since \(r^2 = 9\)).

The equation of the line is:
\[ 4x + 3y - 9 = 0 \]

2. **Find Points of Intersection:**
To find the points where the line intersects the circle, we can substitute \(y\) from the line's equation into the circle's equation. Rearranging the line's equation gives:
\[ 3y = 9 - 4x \quad \Rightarrow \quad y = 3 - \frac{4}{3}x \]

Now substitute \(y\) into the circle's equation:
\[ x^2 + \left(3 - \frac{4}{3}x\right)^2 = 9 \]

Expanding the equation:
\[ x^2 + \left(9 - 8x + \frac{16}{9}x^2\right) = 9 \]
\[ x^2 + 9 - 8x + \frac{16}{9}x^2 = 9 \]
\[ \left(1 + \frac{16}{9}\right)x^2 - 8x = 0 \]
\[ \frac{25}{9}x^2 - 8x = 0 \]

Factoring out \(x\):
\[ x\left(\frac{25}{9}x - 8\right) = 0 \]
This gives us \(x = 0\) or \(\frac{25}{9}x = 8\), leading to:
\[ x = \frac{72}{25} \]

3. **Find Corresponding y-values:**
For \(x = 0\):
\[ y = 3 - \frac{4}{3}(0) = 3 \quad \Rightarrow \quad (0, 3) \]

For \(x = \frac{72}{25}\):
\[ y = 3 - \frac{4}{3}\left(\frac{72}{25}\right) = 3 - \frac{288}{75} = 3 - \frac{96}{25} = \frac{75 - 96}{25} = -\frac{21}{25} \quad \Rightarrow \quad \left(\frac{72}{25}, -\frac{21}{25}\right) \]

4. **Find the Tangents:**
The point of intersection of the tangents can be found using the formula for the chord of contact:
\[ xx_1 + yy_1 = r^2 \]
where \((x_1, y_1)\) are the points of intersection found above.

For the point \((0, 3)\):
\[ 0 \cdot x + 3y = 9 \quad \Rightarrow \quad 3y = 9 \quad \Rightarrow \quad y = 3 \]

For the point \(\left(\frac{72}{25}, -\frac{21}{25}\right)\):
\[ \frac{72}{25}x - \frac{21}{25}y = 9 \]

5. **Finding the Intersection Point:**
The intersection of the tangents can be calculated by solving the equations of the tangents derived from the points of intersection.

After solving, we find that the point of intersection of the tangents is:
\[ (4, 3) \]

### Final Answer:
The point of intersection of the tangents is \((4, 3)\).