Question Statement

Equations of two diameters of a circle are $x - y = 3$ and $3 x + y = 5$ and its radius is 5. Find the equation of the circle.

Background and Explanation

The center of a circle is the point where any two of its diameters intersect. Once the center $(h, k)$ and the radius $r$ are known, the equation of the circle can be determined using the standard form $(x - h)^{2} + (y - k)^{2} = r^{2}$ .

Solution

The intersection of the diameters is the centre of the circle. Thus, to find the centre, we solve the equations of the diameters simultaneously:

$x - y = 3$ $3 x + y = 5$

Step 1: Solve for the center coordinates

Adding both equations (i) and (ii) to eliminate $y$ : $(x - y) + (3 x + y) 4 x x = 3 + 5 = 8 = 2$

Now, substitute $x = 2$ back into equation (i): $2 - y - y y = 3 = 3 - 2 = - 1$

Thus, the centre of the circle is $A (2, - 1)$ .

Step 2: Determine the equation of the circle

We are given that the radius of the circle is $r = 5$ . Using the standard form $(x - h)^{2} + (y - k)^{2} = r^{2}$ with centre $(2, - 1)$ :

$(x - 2)^{2} + (y - (- 1))^{2} (x - 2)^{2} + (y + 1)^{2} = 5^{2} = 25$

Expanding the squares: $(x^{2} - 4 x + 4) + (y^{2} + 2 y + 1) x^{2} + y^{2} - 4 x + 2 y + 5 x^{2} + y^{2} - 4 x + 2 y - 20 = 25 = 25 = 0$

The final equation of the circle is $x^{2} + y^{2} - 4 x + 2 y - 20 = 0$ .

Key Formulas or Methods Used

Intersection of Diameters: The point of intersection of any two diameters of a circle is its center.
Simultaneous Equations: Used to find the point $(x, y)$ where the two lines meet.
Standard Equation of a Circle: $(x - h)^{2} + (y - k)^{2} = r^{2}$ , where $(h, k)$ is the center and $r$ is the radius.
General Equation of a Circle: $x^{2} + y^{2} + 2 g x + 2 f y + c = 0$ .

Summary of Steps

Set up a system of linear equations using the given diameter equations.
Solve the system simultaneously (by addition or substitution) to find the center of the circle.
Identify the radius provided in the problem statement.
Substitute the center coordinates and the radius into the standard circle formula.
Expand and simplify the equation into the general form.

Step 2: Determine the equation of the circle

We are given that the radius of the circle is

r = 5

. Using the standard form

(x - h)^{2} + (y - k)^{2} = r^{2}

with centre

(2, - 1)

(x - 2)^{2} + (y - (- 1))^{2} (x - 2)^{2} + (y + 1)^{2} = 5^{2} = 25

Expanding the squares:

(x^{2} - 4 x + 4) + (y^{2} + 2 y + 1) x^{2} + y^{2} - 4 x + 2 y + 5 x^{2} + y^{2} - 4 x + 2 y - 20 = 25 = 25 = 0

The final equation of the circle is

x^{2} + y^{2} - 4 x + 2 y - 20 = 0

Key Formulas or Methods Used

Intersection of Diameters: The point of intersection of any two diameters of a circle is its center.

Simultaneous Equations: Used to find the point

(x, y)

where the two lines meet.

Standard Equation of a Circle:

(x - h)^{2} + (y - k)^{2} = r^{2}

, where

(h, k)

is the center and

r

is the radius.

General Equation of a Circle:

x^{2} + y^{2} + 2 g x + 2 f y + c = 0

Summary of Steps

Set up a system of linear equations using the given diameter equations.

Solve the system simultaneously (by addition or substitution) to find the center of the circle.

Identify the radius provided in the problem statement.

Substitute the center coordinates and the radius into the standard circle formula.

Expand and simplify the equation into the general form.

7.1 Q-8

Question Statement

Background and Explanation

Solution

Step 1: Solve for the center coordinates

Step 2: Determine the equation of the circle

Key Formulas or Methods Used

Summary of Steps

7.1 Q-8

Question Statement

Background and Explanation

Solution

Step 1: Solve for the center coordinates

Step 2: Determine the equation of the circle

Key Formulas or Methods Used

Summary of Steps