Question

The line y=x+1 intersects the circlex2+y2-4x-2y+10, at points A and B. If C is the centre of the circle, find the: Centre and radius of the circle; Co-ordinates of A and B. The end points of a diameter of a circle are (6, 2) and(4,-3). Find the equation of the circle.​

Answer 1

`(x - 2)^(2) + (y - 1)^(2) = 17`

Explanation:

The current equation of the circle is:

⇒
`x^(2) + y^(2) - 4x - 2y + 10 = 0`

In order to get it into the standard form;

⇒
`(x - a)^(2) + (y - b)^(2) = r^(2)`

We must complete the square;

⇒
`(x - 2)^(2) - 4 + (y - 1) - 1 + 10 = 0`

Now, collect like terms and rearrange;

⇒
`(x - 2)^(2) + (y - 1)^(2) = -5?`

We now know that the Centre is at the point (2, 1).

We can use the distance formula to find the radius;

⇒
`d = \sqrt{(x_(2) - x_(1))^(2) + (y_(2) - y_(1))^(2)}`

⇒
`d = \sqrt{(6 - 2)^(2) + (2 - 1)^(2)}`

⇒
`√(17)`

Therefore the radius squared is 17.

Now substitute into our equation:

⇒
`(x - 2)^(2) + (y - 1)^(2) = 17`