Question

In the given figure, PA is the tangent drawn from an external point P to the circle with center O. If the radius of the circle is 3 cm and PA = 4 cm, then the length of PB is

• A. 3 cm
• B. 4 cm
• C. 5 cm
• D. 2 cm

Answer 1

The Correct Option is D

We can solve this problem using vectors instead of the Pythagorean theorem. The key idea is to consider the vectors in the plane and use the fact that \( \overrightarrow{OA} \) is perpendicular to \( \overrightarrow{PA} \), making triangle \( OAP \) a right triangle.

1. Define vectors:

   Let the vector \( \overrightarrow{OA} \) represent the radius of the circle, which is 3 cm. Let the vector \( \overrightarrow{PA} \) represent the line from point A to P, which is 4 cm. Since \( \overrightarrow{OA} \) is perpendicular to \( \overrightarrow{PA} \), we know that \( \overrightarrow{PA} \) is orthogonal to \( \overrightarrow{OA} \).

2. Compute the length of \( \overrightarrow{OP} \):

   We apply the Pythagorean theorem in terms of vectors. The length of vector \( \overrightarrow{OP} \) is the magnitude of the resultant of the vectors \( \overrightarrow{OA} \) and \( \overrightarrow{PA} \), which are perpendicular:

   \[ |\overrightarrow{OP}| = \sqrt{|\overrightarrow{OA}|^2 + |\overrightarrow{PA}|^2} \]

   Substituting the known values:

   \[ |\overrightarrow{OP}| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \text{ cm}. \]

3. Now compute \( \overrightarrow{BP} \):

   Since \( \overrightarrow{OB} \) is also a radius of the circle and has the same magnitude as \( \overrightarrow{OA} \), we know:

   \[ |\overrightarrow{OB}| = 3 \text{ cm}. \]

   The vector \( \overrightarrow{BP} \) is the difference between \( \overrightarrow{OP} \) and \( \overrightarrow{OB} \), so we have:

   \[ |\overrightarrow{BP}| = |\overrightarrow{OP}| - |\overrightarrow{OB}| = 5 - 3 = 2 \text{ cm}. \]

Final Answer:

The final answer is \( \boxed{2} \) cm.

Answer 2

Given that PA is a tangent to the circle at A, OA is the radius, and O is the center, OA is perpendicular to PA. So, triangle OAP is a right-angled triangle with angle OAP = 90 degrees.

Given OA = 3 cm (radius) and PA = 4 cm.

Using the Pythagorean theorem in triangle OAP, we have:
$OP^2 = OA^2 + PA^2$
$OP^2 = 3^2 + 4^2 = 9 + 16 = 25$
So, $OP = \sqrt{25} = 5$ cm.

We are given that OB is also a radius of the circle, so OB = 3 cm.

Now, we have OP = OB + BP. Therefore,
BP = OP - OB = 5 - 3 = 2 cm.

Final Answer: The final answer is $\boxed{2}$