Question

Find the equation of the circle passing through (0,0) and making intercepts a and b on the coordinate axes.​

Answer 1

Answer:

Step-by-step explanation:

Answer 2

Given:-

• The equation of the circle passing through (0,0).
• The intercepts a and b on the coordinate axes.

To find:-

• Find the equation of the circle .?

Solutions:-

• Let the equation of the required circle be (x - h)² + (y - k)² = r²

Since the circle passes through (0, 0)

=> (0 - h)² + (0 - k)² = r²

=> h² + k² = r²

The equation of the circle.

=> (x - h)² + (y - k)² = h² + k²

Given that, intercepts a and b on the coordinates axis.

The circle passes through points (a, 0) and (0, b).

Therefore,

=> (a - h)² + (0 - k)² = h² + k² .......(i).

=> (0 - h)² + (b - k)² = h² + k² .......(ii).

From equation (i), we obtain.

=> a² - 2ah + h² + k² = h² + k²

=> a² - 2ab = 0

=> a(a - 2h) = 0

=> a = 0 or (a - 2h) = 0

=> a = 0 doesn't equal to zero

Hence, (a - 2h) = 0

=> h = a/2

From equation (ii), we obtain.

=> h² + b² - 2bk + k² = h² + k²

=> b² - 2bk = 0

=> b(b - 2k) = 0

=> b = 0 or (b - 2k) = 0

=> b = 0 (doesn't equal to zero)

Hence, (b - 2k) = 0

=> k = b/2

Thus, the equation of the required Circle.

=> (x - a/2)² + (y - b/2)² = (a/2)² + (b/2)²

=> (2x - a/2)² + (2y - b/2)² = a² + b²/4

=> 4x² + 4y² - 4ax - 4by = 0

=> x² + y² - ax - by = 0

Hence, the equation of the required Circle is x² + y² - ax - by = 0.