Find the equation of the circle which passes through the points (2, 3), (4, 5) and
centre lies on the line y - 4x +3 = 0.
Answers
EXPLANATION.
Equation of the circle which passes through the points,
(2,3) & (4,5).
Centre lies on the line : y - 4x + 3 = 0.
As we know that,
General equation of circle,
⇒ x² + y² + 2gx + 2fy + c = 0.
Centre of circle = (-g,-f).
Radius of the circle = √(g)² + (f)² - c.
Equation of circle passes through point = (2,3).
Put the point in the equation, we get.
⇒ (2)² + (3)² + 2g(2) + 2f(3) + c = 0.
⇒ 4 + 9 + 4g + 6f + c = 0.
⇒ 4g + 6f + 13 + c = 0. ⇒ (1).
Equation of circle passes through point = (4,5).
Put the point in the equation, we get.
⇒ (4)² + (5)² + 2g(4) + 2f(5) + c = 0.
⇒ 16 + 25 + 8g + 10f + c = 0.
⇒ 8g + 10f + 41 + c = 0.
⇒ c = -(8g + 10f + 41). ⇒ (2).
From equation, (1) & (2), we get.
Put the value of equation (2), we get.
⇒ 4g + 6f + 13 - (8g + 10f + 41 ) = 0.
⇒ 4g + 6f + 13 - 8g - 10f - 41 = 0.
⇒ - 4g - 4f - 28 = 0.
⇒ -4(g + f + 7) = 0.
⇒ g + f + 7 = 0. ⇒ (3).
As we know that,
Centre of the Circle = (-g,-f).
Centre lies on the line : y - 4x + 3 = 0.
Put x = -g & y = -f in equation, we get.
⇒ -f -4(-g) + 3 = 0.
⇒ -f + 4g + 3 = 0.
⇒ f = 4g + 3. ⇒ (4).
From equation, (3) &(4), we get.
Put the value of equation (4) in equation (3), we get.
⇒ g + f + 7 = 0.
⇒ g + ( 4g + 3) + 7 = 0.
⇒ g + 4g + 3 + 7 = 0.
⇒ 5g + 10 = 0.
⇒ 5g = -10.
⇒ g = -2.
Put the value of g = -2 in equation (4), we get.
⇒ f = 4g + 3.
⇒ f = 4(-2) + 3.
⇒ f = -8 + 3.
⇒ f = -5.
Put the value of g = -2 & f = -5 in equation (2), we get.
⇒ c = -(8g + 10f + 41).
⇒ c = -[ 8(-2) + 10(-5) + 41 ].
⇒ c = - [ -16 - 50 + 41 ].
⇒ c = + 16 + 50 - 41.
⇒ c = 66 - 41.
⇒ c = 25.
Put the value of g = -2 & f = -5 & c = 25 in general equation of circle.
⇒ x² + y² + 2gx + 2fy + c = 0.
⇒ x² + y² + 2(-2)x + 2(-5)y + 25 = 0.
⇒ x² + y² - 4x - 10y + 25 = 0.
Centre lies on the line y - 4x + 3 = 0
Let x = h
» y = 4 h - 3
So the center is of the form (h, 4h - 3)
Distance of centre from (2,3) and (4,5) will be equal
»(h-2)² + (4h-3 – 3)² = (h-4)² +
(4h-3-5)²
h= 2
So the centre is (2,5)
r=√ (2 – 2)² + (5 – 3)² = 2
So the equation of the circle is
(x - 2)²+ (y – 5)²= 2²
x²+ y² – 4x – 10y + 25 = 0
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