A circle with radius `r` has a chord PQ whose length is `2a`. The tangents drawn at points `P and Q` to the circle meet at `T`, what is the length of `TP`?

Options

A. `(ar)/( sqrt( r^2 - a^2) )`
B. `( 2ar)/( sqrt( r^2 -a^2) ) `
C. `( r^2 + a^2)/( sqrt( r^2 - a^2) )`
D. `( ar)/( r-a)`

Answer

To find the length of \( TP \) where \( T \) is the intersection of the tangents drawn at points \( P \) and \( Q \) on a circle with radius \( r \) and chord \( PQ \) of length \( 2a \), we can follow these steps:

### Step 1: Understand the Geometry
- The chord \( PQ \) has a length of \( 2a \), which means that the distance from the center of the circle \( O \) to the midpoint \( M \) of the chord \( PQ \) can be found using the Pythagorean theorem.
- The radius \( r \) forms a right triangle with half the chord length \( a \) and the distance \( OM \) from the center to the chord.

### Step 2: Calculate \( OM \)
Using the Pythagorean theorem:
\[ OM^2 + a^2 = r^2 \]
Thus,
\[ OM = \sqrt{r^2 - a^2} \]

### Step 3: Analyze the Tangents
- The tangents \( TP \) and \( TQ \) are equal in length because they are drawn from the same external point \( T \) to points \( P \) and \( Q \).
- The angle between the line \( OT \) (from the center to point \( T \)) and the tangent line \( TP \) is \( 90^\circ \).

### Step 4: Use Similar Triangles
- In triangle \( OPT \) (where \( O \) is the center, \( P \) is on the circle, and \( T \) is the point where tangents meet), we can use the properties of similar triangles.
- The triangles \( OPT \) and \( OMT \) are similar because they share angle \( O \) and both have a right angle.

### Step 5: Set Up Ratios
From the similarity of triangles, we can set up the ratio:
\[ \frac{OT}{OM} = \frac{OP}{OP} \]
Where:
- \( OT = TP \)
- \( OM = \sqrt{r^2 - a^2} \)
- \( OP = r \)

Thus, we can express \( TP \):
\[ \frac{TP}{\sqrt{r^2 - a^2}} = \frac{r}{TP} \]

### Step 6: Solve for \( TP \)
Cross-multiplying gives:
\[ TP^2 = r \cdot \sqrt{r^2 - a^2} \]
Taking the square root:
\[ TP = \frac{r^2}{\sqrt{r^2 - a^2}} \]

### Final Answer
The length of \( TP \) is:
\[ TP = \frac{r^2}{\sqrt{r^2 - a^2}} \]