
Find the equation of the director circle of the circle

Answer
528.3k+ views
1 likes
Hint- Here, we will be using the general equation of the director circle to any given circle. The required equation of the director circle is obtained by finding the radius of the given circle and then multiplying it with in order to get the radius of the director circle.
Complete step-by-step answer:
Given equation of circle is
The general equation of any circle having centre coordinates (b,c) and radius r is given by
By comparing equations (1) and (2), we have
The centre coordinates of the given circle is (h,k) and radius of the given circle is a.
As we know that the director circle is the locus of the point of intersection of two perpendicular tangents to the given circle.
For any general circle whose equation is , the equation of director circle is given by .
Clearly, in the case of a director circle the coordinates of the centre of the director circle and the given circle will be the same whereas the radius of the director circle will become times the radius of the given circle.
So, the radius of the director circle for the given circle whose radius is a will become and the centre coordinates of the director circle being the same as that of the i.e., (h,k).
So, the equation of director circle with centre coordinates (h,k) and radius is given by
Hence equation of the director circle of the circle is .
Note- In this problem, the figure is drawn and point P in the figure represents the point of intersection of two perpendicular tangents to the given radius. The locus of this point P will give the director circle to the given circle (any point on the outer circle whose radius is corresponds to the point of intersection of two perpendicular tangents to the smaller circle whose radius is a).
Complete step-by-step answer:
Given equation of circle is
The general equation of any circle having centre coordinates (b,c) and radius r is given by
By comparing equations (1) and (2), we have
The centre coordinates of the given circle is (h,k) and radius of the given circle is a.
As we know that the director circle is the locus of the point of intersection of two perpendicular tangents to the given circle.
For any general circle whose equation is
Clearly, in the case of a director circle the coordinates of the centre of the director circle and the given circle will be the same whereas the radius of the director circle will become
So, the radius of the director circle for the given circle
So, the equation of director circle with centre coordinates (h,k) and radius
Hence equation of the director circle of the circle
Note- In this problem, the figure is drawn and point P in the figure represents the point of intersection of two perpendicular tangents to the given radius. The locus of this point P will give the director circle to the given circle (any point on the outer circle whose radius is
Latest Vedantu courses for you
Grade 10 | MAHARASHTRABOARD | SCHOOL | English
Vedantu 10 Maharashtra Pro Lite (2025-26)
School Full course for MAHARASHTRABOARD students
₹31,500 per year
Recently Updated Pages
Master Class 11 Economics: Engaging Questions & Answers for Success

Master Class 11 Business Studies: Engaging Questions & Answers for Success

Master Class 11 Accountancy: Engaging Questions & Answers for Success

Master Class 11 English: Engaging Questions & Answers for Success

Master Class 11 Computer Science: Engaging Questions & Answers for Success

Master Class 11 Maths: Engaging Questions & Answers for Success

Trending doubts
State and prove Bernoullis theorem class 11 physics CBSE

What are Quantum numbers Explain the quantum number class 11 chemistry CBSE

Write the differences between monocot plants and dicot class 11 biology CBSE

1 ton equals to A 100 kg B 1000 kg C 10 kg D 10000 class 11 physics CBSE

State the laws of reflection of light

One Metric ton is equal to kg A 10000 B 1000 C 100 class 11 physics CBSE
