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JEE • MATHEMATICS • SYSTEM OF CIRCLES

System of circles practice · Mixed difficulty

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Showing 91-98 of 98 questions
91 Easy
91 of 98
Analyzing +4 | -1
The point (3,−4)\left( {3, - 4} \right) lies on both the circles x2+y2−2x+8y+13=0{x^2} + {y^2} - 2x + 8y + 13 = 0 and x2+y2−4x+6y+11=0{x^2} + {y^2} - 4x + 6y + 11 = 0. Then, the angle between the circles is
View answer and explanation
Correct option: D
450{45^0}
Option D (450{45^0}) is correct because it is the angle between the circles, calculated using the formula involving the distance between centers and the radii.
92 Easy
92 of 98
Application +4 | -1
The equation of a circle which cuts the three circles
x2+y2−2  x−6  y+14=0x2+y2−x−4  y+8=0x2+y2+2  x−6  y+9=0\begin{array}{l} {x^2} + {y^2} - 2\;x - 6\;y + 14 = 0\\ {x^2} + {y^2} - x - 4\;y + 8 = 0\\ {x^2} + {y^2} + 2\;x - 6\;y + 9 = 0 \end{array}
orthogonally, is
View answer and explanation
Correct option: A
x2+y2−2  x−4  y+1=0{x^2} + {y^2} - 2\;x - 4\;y + 1 = 0
The correct answer is A (x2+y2−2  x−4  y+1=0{x^2} + {y^2} - 2\;x - 4\;y + 1 = 0) because it represents the circle that cuts the three given circles orthogonally, determined by finding the radical center and the appropriate radius.
93 Easy
93 of 98
Application +4 | -1
Equation of the circle which cuts each of the circles x2+y2+2gx+c=0,x2+y2+2g1x+c=0x^2+y^2+2 g x+c=0, x^2+y^2+2 g_1 x+c=0 and x2+y2+2hx+2ky+a=0x^2+y^2+2 h x+2 k y+a=0 orthogonally is
View answer and explanation
Correct option: B
k(x2+y2)+(a−c)y−ck=0k\left(x^2+y^2\right)+(a-c) y-c k=0
Option B (k(x2+y2)+(a−c)y−ck=0k(x^2+y^2)+(a-c)y-ck=0) is correct because it represents the circle that cuts the given circles orthogonally, derived by setting up equations based on the orthogonality condition.
94 Easy
94 of 98
Application +4 | -1
Two equal circles with their centres on xx and yy axis will possess the radical axis in the following form
View answer and explanation
Correct option: B
2gx−2fy+g2−f2=02 g x-2 f y+g^2-f^2=0
Option B (2gx−2fy+g2−f2=02 g x-2 f y+g^2-f^2=0) is correct because it represents the general form of the radical axis when the centers lie on the x and y axes.
95 Easy
95 of 98
Application +4 | -1
The value of kk so that x2+y2+kx+4y+2=0x^2+y^2+k x+4 y+2=0 and 2(x2+y2)−4x−3y+k=02\left(x^2+y^2\right)-4 x-3 y+k=0 cut orthogonally, is
View answer and explanation
Correct option: C
−103\frac{-10}{3}
Option C (−103\frac{-10}{3}) is correct because the condition for orthogonal intersection of two circles (2(g1g2+f1f2)=c1+c22(g_1g_2 + f_1f_2) = c_1 + c_2) is applied to find the value of kk.
96 Easy
96 of 98
Application +4 | -1
A circle passes through the origin and has its centre on y=xy = x. If it cuts x2+y2−4  x−6  y+10=0{x^2} + {y^2} - 4\;x-6\;y + 10 = 0 orthogonally, then the equation of the circle is
View answer and explanation
Correct option: C
x2+y2−2  x−2  y=0{x^2} + {y^2} - 2\;x - 2\;y = 0
Option C x2+y2−2  x−2  y=0{x^2} + {y^2} - 2\;x - 2\;y = 0 is correct because it satisfies the condition of passing through the origin, having its center on y = x, and cutting the given circle orthogonally.
97 Easy
97 of 98
Understanding +4 | -1
The radical axis of two circles having centres at C1C_1 and C2C_2 and radii r1r_1 and r2r_2 is neither intersecting nor touching any of the circles, if
View answer and explanation
Correct option: B
0<C1C2<∣r1−r2∣0<\mathrm{C}_1 \mathrm{C}_2<\left|r_1-r_2\right|
Option B (0<C1C2<∣r1−r2∣0<\mathrm{C}_1 \mathrm{C}_2<\left|r_1-r_2\right|) is correct because it represents the condition where one circle is completely inside the other, and the radical axis lies outside both circles.
98 Easy
98 of 98
Understanding +4 | -1
The coaxial system of circles given by x2+y2+2gx+c=0{x^2} + {y^2} + 2gx + c = 0 for c<0c < 0 respects
View answer and explanation
Correct option: A
Intersecting circles
Option A (Intersecting circles) is correct because for c<0c < 0, the radical axis has two real points, indicating intersecting circles.
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