Concept:

Distance Formula: 

The distance 'd' between two points (x₁, y₁) and (x₂, y₂) is obtained by using the Pythagoras' Theorem:

• d² = (x₁ - x₂)² + (y₁ - y₂)²

Circle:

The equation of a circle with center (a, b) and radius r is given by:

• (x - a)² + (y - b)² = r²

Calculation:

Let's say that the points on the circle are A(-1, 1) and B(2, 1),

And the center of the circle is a point O(a, b).

Since, The center lies on the line x + 2y + 3 = 0,

The values a and b must satisfy the above linear equation

⇒ a + 2b + 3 = 0       ----(1)

Also, The distance OA = OB = radius of the circle.

Using the distance formula, we get

(a + 1)2 + (b - 1)2 = (a - 2)2 + (b - 1)2

⇒ (a + 1)² - (a - 2)² = 0

⇒ (a + 1 - a + 2)(a + 1 + a - 2) = 0

⇒ (3)(2a - 1) = 0

⇒ \(\rm a = \dfrac12\).

Using equation (1), we get:

\(\rm \dfrac12+2b+3=0\)

⇒ \(\rm b=-\dfrac{7}{4}\).

So, The center of the circle is \(\rm O\left(\dfrac{1}{2},-\dfrac{7}{4} \right)\).

Also, \(\rm r^2=OA^2=\left(\dfrac{1}{2}+1\right)^2+\left(-\dfrac{7}{4}-1\right)^2=\dfrac{9}{4}+\dfrac{121}{16}=\dfrac{157}{16}\).

Now, the equation of the circle will be:

(x - a)2 + (y - b)2 = r2

⇒ \(\rm \left (x - \dfrac12 \right )^2 + \left (y + \dfrac74 \right )^2 = \dfrac{157}{16}\)

⇒ \(\rm x^2 - x+\dfrac14 + y^2 + \dfrac72y+\dfrac{49}{16} = \dfrac{157}{16}\)

⇒ \(\rm x^2 + y^2 + \dfrac{7y}{2}- x+\left (\dfrac{4+49-157}{16} \right ) =0\)

⇒ \(\rm x^2 + y^2 + \dfrac{7y}{2}- x-\dfrac{13}{2} =0\)

∴ The equation of the circle is 2x² + 2y² - 2x + 7y - 13 = 0.