If a chord of contact of tangents drawn from a point P with respect to the circle `x^(2)+y^(2)=9` is x=2, then area, in square units, of triangle formed by tangents drawn from P to the circle and their chord of contact is equal to

Options

A. `(4 sqrt(5))/(2)`
B. `(9sqrt(3))/(2)`
C. `5sqrt(5))/(2)`
D. none of these

Explanation

To solve the problem, we need to find the area of the triangle formed by the tangents drawn from point P to the circle and their chord of contact. The circle is given by the equation \(x^2 + y^2 = 9\), which has a center at the origin (0,0) and a radius of 3.

### Step 1: Identify the chord of contact
The chord of contact from point P to the circle is given as \(x = 2\). This means that the line \(x = 2\) is the chord of contact.

### Step 2: Determine the coordinates of point P
The general equation of the chord of contact from an external point \(P(x_1, y_1)\) with respect to the circle \(x^2 + y^2 = a^2\) is given by:
\[ xx_1 + yy_1 = a^2 \]
For our circle, \(a^2 = 9\). Since the chord of contact is \(x = 2\), we can rewrite it in the form:
\[ x - 2 = 0 \]
Comparing this with the general form, we have:
- \(x_1 = 2\)
- \(y_1 = 0\)

Now, substituting \(x_1\) into the chord of contact equation:
\[ 2x + 0y = 9 \]
This gives us:
\[ x = \frac{9}{2} \]
Thus, the coordinates of point P are \(\left(\frac{9}{2}, 0\right)\).

### Step 3: Find the points of tangency
To find the points of tangency, we substitute \(x = 2\) into the circle's equation:
\[ 2^2 + y^2 = 9 \implies 4 + y^2 = 9 \implies y^2 = 5 \implies y = \pm \sqrt{5} \]
Thus, the points of tangency are \(A(2, \sqrt{5})\) and \(B(2, -\sqrt{5})\).

### Step 4: Calculate the length of segment AB
The length of segment \(AB\) can be calculated as:
\[ AB = |y_1 - y_2| = |\sqrt{5} - (-\sqrt{5})| = 2\sqrt{5} \]

### Step 5: Calculate the perpendicular distance from point P to the chord of contact
The chord of contact is the line \(x = 2\). The distance \(d\) from point \(P\left(\frac{9}{2}, 0\right)\) to the line \(x = 2\) is given by:
\[ d = \left| x_1 - 2 \right| = \left| \frac{9}{2} - 2 \right| = \left| \frac{9}{2} - \frac{4}{2} \right| = \left| \frac{5}{2} \right| = \frac{5}{2} \]

### Step 6: Calculate the area of triangle APB
The area \(A\) of triangle \(APB\) can be calculated using the formula:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]
Here, the base \(AB = 2\sqrt{5}\) and the height \(d = \frac{5}{2}\):
\[ \text{Area} = \frac{1}{2} \times 2\sqrt{5} \times \frac{5}{2} = \frac{5\sqrt{5}}{2} \]

Thus, the area of the triangle formed by the tangents drawn from point P to the circle and their chord of contact is \(\frac{5\sqrt{5}}{2}\) square units.

### Final Answer:
The area of the triangle is \(\frac{5\sqrt{5}}{2}\) square units.

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