The equation of the circle passing through th | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(B) The equation of a circle passing through the intersection of two circles $S_1 = 0$ and $S_2 = 0$ is given by $S_1 + \lambda S_2 = 0$.Here,$S_1 = x

Vedclass · Accepted Answer

$x^2 + y^2 - 3x + 1 = 0$

Vedclass · Answer

(B) The equation of a circle passing through the intersection of two circles $S_1 = 0$ and $S_2 = 0$ is given by $S_1 + \lambda S_2 = 0$.Here,$S_1 = x^2 + y^2 - 6x + 8 = 0$ and $S_2 = x^2 + y^2 - 6 = 0$.The family of circles is $(x^2 + y^2 - 6x + 8) + \lambda(x^2 + y^2 - 6) = 0$.Since the circle passes through the point $(1, 1)$,we substitute $x = 1$ and $y = 1$ into the equation:$(1^2 + 1^2 - 6(1) + 8) + \lambda(1^2 + 1^2 - 6) = 0$$(1 + 1 - 6 + 8) + \lambda(1 + 1 - 6) = 0$$4 + \lambda(-4) = 0$$4 - 4\lambda = 0 \implies \lambda = 1$.Substituting $\lambda = 1$ back into the family equation:$(x^2 + y^2 - 6x + 8) + 1(x^2 + y^2 - 6) = 0$$2x^2 + 2y^2 - 6x + 2 = 0$Dividing by $2$,we get $x^2 + y^2 - 3x + 1 = 0$.

The equation of the circle passing through the points of intersection of the circles $x^2 + y^2 - 6x + 8 = 0$ and $x^2 + y^2 - 6 = 0$ and the point $(1, 1)$ is:

Similar Questions

If the circle $x^2+y^2+8x-4y+c=0$ touches the circle $x^2+y^2+2x+4y-11=0$ externally and cuts the circle $x^2+y^2-6x+8y+k=0$ orthogonally,then $k$ is equal to

If the two circles $2x^2 + 2y^2 - 3x + 6y + k = 0$ and $x^2 + y^2 - 4x + 10y + 16 = 0$ intersect orthogonally,then the value of $k$ is:

The condition for the coaxial system $x^2+y^2+2 \lambda x+c=0$,where $\lambda$ is a parameter and $c$ is a constant,to have distinct limiting points,is

The equation of the family of circles having the same radical axis as the circles $x^2 + y^2 = 4$ and $x^2 + y^2 + 2x + 4y = 6$ is:

The point/points of intersection of the common tangents of the two circles $x^2+y^2-8x-6y+21=0$ and $x^2+y^2-2y-15=0$ is/are

Vedclass Test Series

Exam Paper Generator

Online Exam Module