Q.Find the equation of a circle whose centre is $(3, -1)$ and which cuts off a chord of length 6 units on the line $2x - 5y + 18 = 0$.

The core idea here is to use the standard form of a circle's equation $(x-h)^2 + (y-k)^2 = r^2$, where $(h, k)$ is the center and $r$ is the radius. We are given the center, so we need to find the radius $r$.

1. The center of the circle is given as $(h, k) = (3, -1)$.
2. The perpendicular distance $d$ from the center $(3, -1)$ to the chord line $2x - 5y + 18 = 0$ is calculated using the distance formula:

3. The chord length is 6 units, so half the chord length, $L$, is $6/2 = 3$ units. …

To find the equation of a circle, we generally need two pieces of information: its center $(h, k)$ and its radius $r$. The standard form of a circle's equation is $(x-h)^2 + (y-k)^2 = r^2$.

In this problem, the center is already given as $(3, -1)$. So, we have $h=3$ and $k=-1$. Our main task is to determine the radius $r$.

The key insight comes from the geometry of a circle and its chords. A fundamental property is that the perpendicular drawn from the center of a circle to a chord bisects the chord. This creates a right-angled triangle where:

• The hypotenuse is the radius ($r$) of the circle.
• One leg is half the length of the chord.
• The other leg is the perpendicular distance ($d$) from the center of the circle to the chord.

If we can find this perpendicular distance $d$ and we know half the chord length, we can use the Pythagorean theorem ($r^2 = d^2 + (\text{half chord})^2$) to find $r^2$, and thus write the circle's equation.

Let's break this down into steps:

1. Identify the given information.

   The center of the circle is $C = (h, k) = (3, -1)$.

   The length of the chord is $L = 6$ units.

   The equation of the line containing the chord is $2x - 5y + 18 = 0$.

2. Calculate half the chord length.

   Since the perpendicular from the center bisects the chord, we need half its length for our right-angled triangle.

   Half chord length $= \frac{L}{2} = \frac{6}{2} = 3$ units.

3. Calculate the perpendicular distance from the center to the chord.

   This is the distance $d$ from the point $(3, -1)$ to the line $2x - 5y + 18 = 0$.

   The perpendicular distance $d$ from a point $(x_1, y_1)$ to a line $Ax + By + C = 0$ is given by:

   $d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}}$

   Here, $(x_1, y_1) = (3, -1)$ and the line is $2x - 5y + 18 = 0$, so $A=2$, $B=-5$, and $C=18$.

   Substitute these values into the formula:

$$d = \sqrt{29}$$ …