CameraIcon
CameraIcon
SearchIcon
MyQuestionIcon
MyQuestionIcon
1
You visited us 1 times! Enjoying our articles? Unlock Full Access!
Question

Consider the circles S1:x2+y2+2x+4y+1=0,S2:x2+y24x+3=0&S3:x2+y2+6y+5=0 which of this following statements are correct ?

A
Radical centre of S1,S2 and S3 lies in 1st quadrant
No worries! We‘ve got your back. Try BYJU‘S free classes today!
B
Radical centre of S1,S2 and S3 lies in 4th quadrant
Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses
C
Radius of the circle intersecting S1,S2 and S3 orthogonally is 1.
Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses
D
Circle orthogonal to S1,S2 and S3 has its x and y intercept equal to zero.
No worries! We‘ve got your back. Try BYJU‘S free classes today!
Open in App
Solution

The correct options are
B Radical centre of S1,S2 and S3 lies in 4th quadrant
C Radius of the circle intersecting S1,S2 and S3 orthogonally is 1.
Radical axes of two circles S1 and S2 is given by S1S2=0.
So, for given circles radical axes are,
6x+4y2=0...(1)
2y2x+4=0...(2)
4x+6y+2=0...(3)

Let's find the radical center by finding intersection of the lines,
(1) - 3(2)
10y=10y=1

Now,
6x+4y2=0
6x42=0
x=1

Hence, the radical center is (1.-1).
So, the option b is correct answer.

We know that radical center is center of the circle orthogona to three circles.

And also we know that length of tangent from center of orthogonal circle to any of parent circle is radius of orthogonal circle.

Now length of tangent from pont (h,k) to circle x2+y2+2gx+2fy+c=0 is given by
h2+k2+2gh+2fk+c=0

So, length of tangent from center of orthogonal circle i.e., (1,-1) to circle x2+y2+2x+4y+1=0 is,
=12+(1)2+2(1)+4(1)+1
=1
= Radius of orthogonal circle
So, C is also the correct answer.

Now, we know that the general equation of circle with center (h,k) and radius r is,
x2+y2+2gx+2fy+c=0

So, for orthogonal circle, equation is,
(x1)2+(y+1)2=1

So, the intersection points of circle with x-axis are (2,0),(0,0)xintercept=2
And, the intersection points of circle with y-axis are (0,0),(0,2)yintercept=2

Hence, (B) and (C) are the correct answer.


flag
Suggest Corrections
thumbs-up
0
Join BYJU'S Learning Program
similar_icon
Related Videos
thumbnail
lock
Lines and Points
MATHEMATICS
Watch in App
Join BYJU'S Learning Program
CrossIcon