Write the equation of a circle with a center at (1, 4) where a point on the circle is (4, 8).
Solution:
Given: Center of the circle = (1, 4)
A point on the circle = (4, 8)
The general equation of a circle is given by (x - α)2 +(y -β)2 = r2 --- (1)
Where (x, y) is any point on the circle, (α, β) is the center of the given circle, and 'r' is the radius of the given circle.
To find the radius we will use the distance formula (x\(_1\) - x\(_2\))2 + (y\(_1\) - y\(_2\))2 = r2
(x\(_1\), y\(_1\)) = (4, 8) and (x\(_2\) , y\(_2\)) = (1, 4)
⇒ (4 - 1)2 + (8 - 4)2 = r2
⇒ 32 + 42 = r2
⇒ 9 + 16 = r2
⇒ 25 = r2
⇒ √25 = √r2
⇒ 5 = r
Therefore, the radius = 5 units.
Now we will use eq(1) to find the equation of the circle
⇒ (x - 1)2 + (y - 4)2 = 52
Expanding the above equation by using the identities
(x + y)2 = x2 + y2 + 2xy
(x - y)2 = x2 + y2 - 2xy
⇒ x2 + 1 - 2x + y2 + 42 - 2.4.y = 52
⇒ x2+ y2 - 2x - 8y + 1 + 16 = 25
⇒ x2 + y2 - 2x - 8y + 17 - 25 = 0
⇒ x2 + y2 - 2x - 8y - 8 = 0
Therefore, the equation of circle is x2 + y2 - 2x - 8y - 8 = 0.
Write the equation of a circle with a center at (1, 4) where a point on the circle is (4, 8).
Summary:
The equation of the circle having the center at (1, 4) and a point on the circle at (4, 8) is x2 + y2 - 2x - 8y - 8 = 0.