The radius of the circle which cuts the circl | vedclass

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

Question

(A) Let the equation of the required circle be $x^2+y^2+2gx+2fy+c=0$. Since it cuts the given circles orthogonally,we use the condition $2g_1g_2 + 2f_

Vedclass · Accepted Answer

$\frac{\sqrt{193}}{4 \sqrt{2}}$

Vedclass · Answer

(A) Let the equation of the required circle be $x^2+y^2+2gx+2fy+c=0$. Since it cuts the given circles orthogonally,we use the condition $2g_1g_2 + 2f_1f_2 = c_1 + c_2$. For $x^2+y^2-4x-4y+7=0$: $2g(-2) + 2f(-2) = c+7$ $\Rightarrow -4g-4f-c=7$ $\Rightarrow 4g+4f+c=-7$ $(i)$. For $x^2+y^2+4x-4y+6=0$: $2g(2) + 2f(-2) = c+6 \Rightarrow 4g-4f-c=6$ $(ii)$. For $x^2+y^2+4x+4y+5=0$: $2g(2) + 2f(2) = c+5 \Rightarrow 4g+4f-c=5$ $(iii)$. Subtracting $(iii)$ from $(i)$: $(4g+4f+c) - (4g+4f-c) = -7-5$ $\Rightarrow 2c = -12$ $\Rightarrow c = -6$. Substituting $c=-6$ into $(i)$ and $(ii)$: $4g+4f = -1$ and $4g-4f = 0$. Adding these: $8g = -1 \Rightarrow g = -\frac{1}{8}$. Subtracting: $8f = -1 \Rightarrow f = -\frac{1}{8}$. The radius $r = \sqrt{g^2+f^2-c} = \sqrt{(-\frac{1}{8})^2 + (-\frac{1}{8})^2 - (-6)} = \sqrt{\frac{1}{64} + \frac{1}{64} + 6} = \sqrt{\frac{2}{64} + 6} = \sqrt{\frac{1}{32} + 6} = \sqrt{\frac{193}{32}} = \frac{\sqrt{193}}{4\sqrt{2}}$.

The radius of the circle which cuts the circles $x^2+y^2-4x-4y+7=0$,$x^2+y^2+4x-4y+6=0$,and $x^2+y^2+4x+4y+5=0$ orthogonally is

Similar Questions

If the circle $S \equiv x^2+y^2+2gx+4y+1=0$ bisects the circumference of the circle $x^2+y^2-2x-3=0$,then the radius of circle $S=0$ is

The radical axis of the co-axial system of circles with limiting points $(1, 2)$ and $(-2, 1)$ is

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:

If the circles $x^2+y^2-6x-8y-12=0$ and $x^2+y^2-4x+6y+k=0$ are orthogonal to each other,then the value of $k$ is:

Vedclass Test Series

Exam Paper Generator

Online Exam Module