Question

A circle has radius equal to 3 units and its centre lies on the line y = x – 1. Find the equation of the circle if it passes through (7, 3).❤​

Answer 1

▄▄▄▄▄▄

▌Given  ▌

• A circle has radius equal to 3 units, r = 3 units.
• Its centre lies on the line y = x - 1.

▄▄▄▄▄▄▄

▌To Find  ▌

The equation of the circle if it passes through (7, 3).

▄▄▄▄▄▄▄

▌Solution ▌

Let the centre of the circle be (a, b). It lies on the line y = x - 1.

⇒ β = α - 1

As the centre is (α, α - 1).

The equation of the circle is :

⇒ (x - α)  + 2(y - α + 1)² = 9

It passes through (7, 3) :

⇒ 2α² - 22α + 56 = 0

⇒ (7 - α)² + (4 - α)² = 9

⇒ α² - 11α + 28 = 0

⇒ α² - 4α - 7α + 28 = 0

⇒ α(α - 4) - 7(α - 4) = 0

⇒ (α - 4)(α - 7) = 0

⇒ α = 4, 7

_______________________

Hence the required equations are

x² + y² - 8x - 6y + 16 = 0  

AND

x² + y² - 14x - 12y + 76 = 0

Answer 2

$\bf{\red{Given}}$

A circle has radius equal to 3 units, r = 3 units.

Its centre lies on the line y = x - 1.

$\bf{\red{To\:Find}}$

The equation of the circle if it passes through (7, 3).

$\bf{\red{Solution}}$

Let the centre of the circle be (a, b). It lies on the line y = x - 1.

⇒ β = α - 1

As the centre is (α, α - 1).

The equation of the circle is :

⇒ (x - α)  + 2(y - α + 1)² = 9

It passes through (7, 3) :

⇒ 2α² - 22α + 56 = 0

⇒ (7 - α)² + (4 - α)² = 9

⇒ α² - 11α + 28 = 0

⇒ α² - 4α - 7α + 28 = 0

⇒ α(α - 4) - 7(α - 4) = 0

⇒ (α - 4)(α - 7) = 0

⇒ α = 4, 7

_______________________

Hence the required equations are

x² + y² - 8x - 6y + 16 = 0  

AND

x² + y² - 14x - 12y + 76 = 0