Question

If two diameters of a circle lie along the lines x - y = 5 and x + y = 7 , and the area of the circle is 50π sq units, find the equation of the circle.
1. \(\rm (x-6)^2+(y-1)^2=50\)
2. \(\rm (x-1)^2+(y-6)^2=16\)
3. \(\rm (x-5)^2+(y-7)^2=12\)
4. \(\rm (x-1)^2+(y-4)^2=25\)

Answer 1

Correct Answer - Option 1 : \(\rm (x-6)^2+(y-1)^2=50\)

Concept:

The general equation for a circle is (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius.

Area of circle = π r² , where, r = radius of circle.

 

Calculation:

Given equations of lines are  x - y = 5 and x + y = 7,  and area of circle = 50π sq units

We know that the point of intersection of two diameters of a circle is the centre of the circle.

On adding  x - y = 5 and x + y = 7, we get 

2x = 12

⇒ x = 6

On putting x = 6 in x - y = 5, we get,

6 - y = 5

⇒ y = 1

∴ Centre of a circle = (6, 1)

 

Now, Area of circle = π r2 =  50π 

⇒ r² = 50

⇒ r = 5√2 units.

 

We have, centre of a circle = (6, 1) and radius = 5√2 units.

∴ The equation of circle is \(\rm (x-6)^2+(y-1)^2=50\)  

Hence, option (1) is correct.