Question

A circle passes through the points (2 3) and (4 5) if its centre lies on the line y-4x+3=0 then its radius is

Answer 1

let equation of circle is x² + y² + 2gx + 2fy + c = 0 where (-g, -f) is the centre of circle
a/c to question, circle passes through the points (2, 3) and (4, 5)

so, 2² + 3² + 2g(2) + 2f(3) + c = 0
4g + 6f + c + 13 = 0 .......(1)

4² + 5² + 2g(4) + 2f(5) + c = 0
8g + 10f + c + 41 = 0 .........(2)

solve equations (1) and (2),
4g + 4f + 28 = 0
g + f + 7 = 0 ..........(3)


centre of circle lies on the line y - 4x + 3 = 0.
so, (-g, -f) satisfies the equation of circle .
e.g., -f +4g + 3 = 0.......(4)

solve equations (3) and (4),
5g + 10 = 0, g = -2 and f = -7 - g = -5
hence, centre is (-2, -5) and c = 25
so, radius of circle is √(g² + f² - c)
= √(2² + 5² - 25) = 2

hence, radius of circle is 2unit.

Answer 2

Answer:

2

Step-by-step explanation:

Hi,

Let the center of the circle be (h, k).

Given that center lies on the line y - 4x + 3 = 0

⇒ k = 4h - 3

Hence , center  O ( h, 4h - 3)

Given that circle passes through A(2, 3) and B(4, 5)

Since O is the center and A and B are points on circle, if r is the

radius of the circle then OA = OB = r

⇒OA² = OB²

⇒(h - 2)² + (4h - 6)² = (h - 4)² + (4h - 8)²

⇒ (h - 2)² - (h - 4)² = (4h - 8)² - (4h - 6)²

⇒2(2h - 6) = -2(8h - 14) [ Using a² - b² = (a - b)(a + b)]

⇒2h - 6 = -8h + 14

⇒ h = 2

Hence, center of circle O ( 2, 5)

Radius of the circle OA = 2 units

Hope, it helped !