System of circles practice · Mixed difficulty

Option D $\left( {1,\; - 4} \right)$ is correct because it satisfies the equation of the common chord obtained by subtracting the equations of the two circles.

Option C (5) is correct because the number of radical axes ($^nC_2$) equals the number of radical centers ($^nC_3$) when $n=5$.

Option C ($(-1,1)(1 / 5,8 / 5)$) is correct because these are the limiting points of the coaxial system containing the two given circles.

Option C (3) is correct because the radius is calculated using the orthogonality condition for circles: $(C_1C_2)^2 = r_1^2 + r_2^2$, where $C_1$ and $C_2$ are the centers of the circles.

The correct answer is $c > 0$ because the condition for distinct limiting points in a coaxial system requires the radius of the circles to be real, which implies that $c$ must be positive.

Option C (${x^2} + {y^2} - 6x - 6y + 9 = 0$) is correct because it satisfies the orthogonality condition, passes through (3, 0), and touches the y-axis.

Option D ($- 23$) is correct because it is derived from the condition that the first circle bisects the circumference of the second circle, implying that the common chord passes through the center of the second circle.

Option D (${y^2} = 16x$) is correct because it represents the locus of the center of the circle that cuts the given circle orthogonally and touches the line x=2, derived by eliminating the constant term and finding the relationship between g and f.

Option C ($9 {x}-10 {y}+11=0$) is correct because it is the equation of the line obtained by subtracting the equations of the two circles after applying the condition for orthogonality.

Option D (-3, 6) is correct because it satisfies the equation of the circle formed by the intersection of the given circles and having its center on the given line.

Option B (orthocentre of the triangle) is correct because the common chords are the altitudes of the triangle, which intersect at the orthocenter.

Option A (3) is correct because the radius is calculated using the condition for orthogonal intersection of two circles.

Option B ($8 x-12 y+5=0$) is correct because it represents the locus of the center of the circle that cuts the given circles orthogonally, derived from the orthogonality condition.

Option D ($x-y-1=0$) is correct because it represents the equation obtained by subtracting the equations of the two circles, which gives the common chord.

The correct answer is A (${x^2} + {y^2} - x - y = 0$) because it represents the equation of a circle with the intercept AB as its diameter, where A and B are the intersection points of the line and the given circle.

Option D $\left( {1,\; - 4} \right)$ is correct because it is the only point that satisfies the equation of the common chord, which is found by subtracting the equations of the two circles.

Option A ($x = 3$) is correct because the equation of the common tangent is found by subtracting the equations of the two circles, ${S_1} - {S_2} = 0$.

Option C ($aa' + bb' = \frac{c}{2} + c'$) is correct because the condition for orthogonal intersection of two circles ($2(g_1g_2 + f_1f_2) = c_1 + c_2$) is applied after adjusting the second equation.

Option B ($\left( {0, - 3} \right)$ and $\left( { - 2, - 1} \right)$) is correct because these are the limiting points found by setting the radius of the coaxial system of circles to zero.

Option C ($- \frac{{10}}{3}$) is correct because it is the value of k that satisfies the condition for two circles to cut orthogonally.

Option C (Cut each other orthogonally) is correct because the condition $2(g_1g_2 + f_1f_2) = c_1 + c_2$ is satisfied, indicating that the circles intersect at right angles.

Option B ($4{x^2} + 4{y^2} + 30x - 13y - 25 = 0$) is correct because it represents the equation of the circle passing through the intersection points of the two given circles and the point (1,1).

Option B (${x^2} + {y^2} - 2x - 8 = 0$) is the correct equation of the circle on PQ as a diameter, found by setting $\lambda = 0$ in the given equation.

The correct answer is (-1/2, -1/2) because it is the midpoint of the chord formed by the intersection of the circle and the line.

Option A (${x^2} + {y^2} - 2y - 6 = 0$) is correct because it represents a circle with AB as a diameter, derived from the intersection of the given line and circle.

Option C ($\frac{{ - 5}}{4}$) is correct because it satisfies the orthogonality condition $2g_1g_2 + 2f_1f_2 = c_1 + c_2$ for the given circles.

Option B ($2 a x+2 b y-\left(a^{2}+b^{2}+4\right)=0$) is correct because it represents the locus of the center of a circle that passes through a given point and cuts another circle orthogonally.

Option D (None of these) is correct because the calculated radical center does not match any of the given options.

Option A ($x^{2}+y^{2}+3 x+y-11=0$) is correct because it represents the equation of the circle formed with the given chord as diameter, found by using the family of circles equation.

Option A (3) is correct because by finding the equation of the common chord and using the fact that it passes through the center of the second circle, we can solve for the radius r.