(D) Two circles $x^2 + y^2 + 2g_1x + 2f_1y + c_1 = 0$ and $x^2 + y^2 + 2g_2x + 2f_2y + c_2 = 0$ intersect orthogonally if $2(g_1g_2 + f_1f_2) = c_1 +

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(D) Two circles $x^2 + y^2 + 2g_1x + 2f_1y + c_1 = 0$ and $x^2 + y^2 + 2g_2x + 2f_2y + c_2 = 0$ intersect orthogonally if $2(g_1g_2 + f_1f_2) = c_1 + c_2$.For the given circles:Circle $1$: $g_1 = \lambda, f_1 = 3, c_1 = 1$Circle $2$: $g_2 = 2, f_2 = 1, c_2 = 0$Substituting these values into the condition:$2(\lambda \times 2 + 3 \times 1) = 1 + 0$$2(2\lambda + 3) = 1$$4\lambda + 6 = 1$$4\lambda = -5$$\lambda = \frac{-5}{4}$.

Class 11 Mathematics 10-1.Circle and System of Circles System of circles

Medium

The value of $\lambda$,for which the circle $x^2 + y^2 + 2\lambda x + 6y + 1 = 0$ intersects the circle $x^2 + y^2 + 4x + 2y = 0$ orthogonally is

A
$\frac{-5}{2}$
B
$-1$
C
$\frac{-11}{8}$
D
$\frac{-5}{4}$

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10-1.Circle and System of Circles

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System of circles

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 $5g+2f+c=$

MediumTS EAMCET 2019

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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

DifficultAP EAMCET 2007

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If one of the diameters of the circle,given by the equation $x^2 + y^2 - 4x + 6y - 12 = 0$,is a chord of a circle $S$ whose center is at $(-3, 2)$,then the radius of $S$ is:

MediumJEE MAIN 2016

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The point $(2, 3)$ is a limiting point of a coaxial system of circles of which $x^2 + y^2 = 9$ is a member. The coordinates of the other limiting point are given by

Difficult

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The limiting points of the co-axial system containing the two circles $x^2+y^2+2x-2y+2=0$ and $25(x^2+y^2)-10x-80y+65=0$ are

MediumAP EAMCET 2001

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The value of $\lambda$,for which the circle $x^2 + y^2 + 2\lambda x + 6y + 1 = 0$ intersects the circle $x^2 + y^2 + 4x + 2y = 0$ orthogonally is

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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 $5g+2f+c=$

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