If 2x+y=0 is the equation of a chord of the c | vedclass

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

Question

(D) The equation of a family of circles passing through the intersection of the circle $S \equiv x^2+y^2-2x-6y+3=0$ and the line $L \equiv 2x+y=0$ is

Vedclass · Accepted Answer

$(-2, 1)$

Vedclass · Answer

(D) The equation of a family of circles passing through the intersection of the circle $S \equiv x^2+y^2-2x-6y+3=0$ and the line $L \equiv 2x+y=0$ is given by $S + \lambda L = 0$. $x^2+y^2-2x-6y+3 + \lambda(2x+y) = 0$ $x^2+y^2 + x(2\lambda-2) + y(\lambda-6) + 3 = 0$. Since the chord $2x+y=0$ is the diameter of this new circle,the center of the circle must lie on the line $2x+y=0$. The center of the circle is $(1-\lambda, \frac{6-\lambda}{2})$. Substituting the center into the line equation: $2(1-\lambda) + \frac{6-\lambda}{2} = 0$. $4 - 4\lambda + 6 - \lambda = 0$ $\Rightarrow 5\lambda = 10$ $\Rightarrow \lambda = 2$. Substituting $\lambda = 2$ into the circle equation: $x^2+y^2 + x(4-2) + y(2-6) + 3 = 0$. $x^2+y^2+2x-4y+3 = 0$. Checking the options,for point $(-2, 1)$: $(-2)^2 + (1)^2 + 2(-2) - 4(1) + 3 = 4 + 1 - 4 - 4 + 3 = 0$. Thus,the circle passes through the point $(-2, 1)$.

If $2x+y=0$ is the equation of a chord of the circle $x^2+y^2-2x-6y+3=0$,then the circle with this chord as diameter passes through the point

Similar Questions

If the equation of the circle which passes through the point $(1,1)$ and cuts both the circles $x^2+y^2-4x-6y+4=0$ and $x^2+y^2+6x-4y+15=0$ orthogonally is $x^2+y^2+2gx+2fy+c=0$,then $5g+2f+c=$

If the circle $x^2+y^2+4x-6y+c=0$ bisects the circumference of the circle $x^2+y^2-6x+4y-12=0$,then $c$ is equal to

If $y = 2x$ is a chord of the circle $x^{2} + y^{2} = 10x$,then find the equation of the circle having this chord as its diameter.

The number of common tangents to the two circles $x^2+y^2-8x+2y=0$ and $x^2+y^2-2x-16y+25=0$ is:

For what value of $k$ do the circles $x^2 + y^2 + 5x + 3y + 7 = 0$ and $x^2 + y^2 - 8x + 6y + k = 0$ intersect orthogonally?

Vedclass Test Series

Exam Paper Generator

Online Exam Module