Question

The centre of a circle is (2x-1, 3x + 1) and radius is 10 units. Find the value of x if the circle passes through the point (-3,-1).​

Answer 1

Given:-

The centre of a circle is (2x-1, 3x + 1)

Radius is 10 units.

To find :-

The value of x if the circle passes through the point (-3,-1).

Solution :-

Given that

The centre of a circle = (2x-1,3x+1)

Radius of the circle = 10 units

Given point = (-3,-1)

We know that

The distance between the centre of the circles and any point on the circle is the radius of the circle.

Therefore, The distance between (2x-1,3x+1) and

(-3,-1) is 10 units

We know that

The distance between two points (x1, y1) and (x2, y2) is √[(x2-x1)²+(y2-y1)²] units

We have,

(x1, y1) = (2x-1,3x+1) => x1 = 2x-1 and

y1 = 3x+1

(x2, y2) = (-3,-1) => x2 = -3 and y2 = -1

Now,

√[(-3-(2x-1))²+(-1-(3x+1))²] = 10

=> √[(-3-2x+1)²+(-1-3x-1)²] = 10

=> √[(-2-2x)²+(-2-3x)²] = 10

=> √[(-(2+2x))²+(-(2+3x))²] = 10

=> √[(2+2x)²+(2+3x)²] = 10

=> √[2²+(2x)²+2(2)(2x)+2²+(3x)²+2(2)(3x)]

= 10

Since, (a+b)² = a²+2ab+b²

=> √(4+4x²+8x+4+9x²+12x) = 10

=> √(13x²+20x+8) = 10

On squaring both sides then

=> [√(13x²+20x+8)]² = 10²

=> 13x²+20x+8 = 100

=> 13x²+20x+8-100 = 0

=> 13x²+20x-92 = 0

=> 13x²-26x+46x-92 = 0

=> 13x(x-2)+46(x-2) = 0

=> (x-2)(13x+46) = 0

=> x-2 = 0 or 13x+46 = 0

=> x = 2 or 13x = -46

=> x = 2 or x = -46/13

Therefore, x = 2 and -46/13

Answer :-

The values of x 2 and -46/13

Used Concept:-

→ The distance between the centre of the circles and any point on the circle is the radius of the circle.

Used formulae:-

→ The distance between two points

(x1, y1) and (x2, y2) is √[(x2-x1)²+(y2-y1)²] units

→ (a+b)² = a²+2ab+b²

Answer 2

Step-by-step explanation:

Given that the circle has centre O(2x - 1, 3x + 1) and passes through the point A(- 3, - 1) and has a radius(r) of 10 units.

We know that the radius of the circle is the distance between the centre and any point on the circle.

So, we have r = OA

⇒ OA = 10

⇒ OA2 = 100

⇒ (2x - 1 - (- 3))2 + (3x + 1 - (- 1))2 = 100

⇒ (2x + 2)2 + (3x + 2)2 = 100

⇒ 4x2 + 8x + 4 + 9x2 + 12x + 4 = 100

⇒ 13x2 + 20x - 92 = 0

⇒ 13x2 - 26x + 46x - 92 = 0

⇒ 13x(x - 2) + 46(x - 2) = 0

⇒ (13x + 46)(x - 2) = 0

⇒ 13x + 46 = 0 (or) x - 2 = 0

⇒ 13x = - 46 (or) x = 2

⇒ x = -46/13 (or) x = 2

∴ The values of the x are -46/13 or 2.