Find the equation of the circle passing through the points (4,1) and (6,5) and whose centre is on the line 4x+y=16

Question

Find the equation of the circle passing through the points (4,1) and (6,5) and whose centre is on the line 4x+y=16

Filo · Accepted Answer

Let the equation of circle be x2+y2+2gx+2fy+c=0        .....(1)where (−g,−f) is center and g2+f2−c is the radius.Since, the circle passes through (4,1), so it satisfies eqn (1)⇒16+1+8g+2f+c=0⇒8g+2f+c=−17      .....(2)Since, the circle passes through (6,5), so it satisfies eqn (1)⇒36+25+12g+10f+c=0⇒12g+10f+c=−61      .....(3)Subtracting eqn (3) from eqn (2), we get−4g−8f=44⇒g+2f=−11        ....(4)Given center lies on the line 4x+y=16Since, center (−g,−f) satisfies this equation.⇒−4g−f=16        ....(5)Solving eqn (4) and (5), we getg=−3,f=−4Put this value in (2), we getc=15Substituting these values in (1),x2+y2−6x−8y+15=0which is the equation of required circle.

Question Text	Find the equation of the circle passing through the points $(4, 1)$ and $(6, 5)$ and whose centre is on the line $4 x + y = 16$
Answer Type	Text solution:1
Upvotes	150

Find the equation of the circle passing through the points $(4, 1)$ and $(6, 5)$ and whose centre is on the line $4 x + y = 16$

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