Question

What are the coordinates of the center of the circle (x-1)²+(y-2)²=2² and y=2x+2?

Answer 1

Answer:

Compare the given equation of the circle (x - 1)² + (y -2)² = 2²

with standard form of circle: (x - h)² + (y - k)² = r²

Here, (h, k) is the center of the circle

and r is the radius of the circle.

Thus, The center of the circle is: (1, 2)

Also, for finding the point of intersections of (x - 1)² + (y -2)² = 2² and y = 2x + 2,

Substitute the value of y from equation of line in the equation of circle.

(x - 1)² + (2x + 2 - 2)² = 2²

⇒ (x - 1)² + (2x)² = 2²

⇒ x² + 1 - 2x + 4x² = 4

⇒ 5x² - 2x - 3 = 0

Applying Middle term splitting method

5x² - 5x + 3x - 3 = 0

⇒ 5x(x - 1) + 3(x - 1) = 0

⇒ (5x + 3)(x - 1) = 0

⇒ x = $\frac{-3}{5}$ and x = 1

Thus, we get coordinates: $(\frac{-3}{5} ,\frac{4}{5} )$ and (1, 4)