Circles

To solve the problem, we need to find the centre of a circle when the endpoints of its diameter are given as (1, 2) and (7, –4).

1. Understanding the Concept:
The centre of a circle is the midpoint of the diameter. If the endpoints of the diameter are and , then the midpoint is given by:

2. Substitute the values:
Given points: and

3. Calculate the Coordinates of the Centre:
The centre of the circle is at .

Final Answer:
The correct answer is option (B): (4, –1).

To solve the problem, we need to find the length of given that , , and . Here's the step-by-step solution:

Step 1: Understand the geometry.
The point is the center of the circle, and is the radius of the circle. Since , is the midpoint of . This means , so .

Step 2: Use the Pythagorean theorem in .
In , is the hypotenuse (which is the radius of the circle, ), , and is the unknown side. By the Pythagorean theorem:

Substituting the known values:

Simplify:

Solve for :

Take the square root of both sides:

Step 3: Find the length of .
Since is the midpoint of , we have:

Final Answer:
The length of is .

To solve the problem, we need to find the length of the tangent segment PB drawn from an external point P to a circle, given that PA is another tangent from the same point and its length is 10 cm.

1. Understanding the Property of Tangents from a Point:
If two tangents are drawn to a circle from the same external point, then the lengths of those tangents are equal.
This means:

2. Using the Given Information:
We are given that:

3. Applying the Tangent Length Property:
Since , we substitute:

Final Answer:
The length of PB is .