Question

 

Find the equation of a circle having the lines x2 + 2xy + 3x + 6y = 0 as its normals and having size just sufficient to contain the circle

                         x(x – 4) + y(y – 3) = 0. 

Solution

Correct option is

 

Given pair of normals is

                  x2 + 2xy + 3x + 6y = 0

or              (x + 2y) (x + 3) = 0 

∴ Normals are x + 2y = 0 and x + 3 = 0 the point of intersection of normals x + 2y = 0 and x + 3 = 0 is the centre of required circle, we get centre C1 ≡ (–3, 3/2) and other circle is

               x (x – 4) + y(y – 3) = 0 

or           x2 + y2 – 4x – 3y = 0             …(1)

  

Since the required circle just contains the given circle (1), the given circle should touch the required circle internally from inside.

  

                                                              

                                                              

Hence equation of required circle is

                  

 

SIMILAR QUESTIONS

Q1

Determine the radius of the circle, two of whose tangents are the lines 2x+ 3y – 9 = 0 and 4x + 6y + 19 = 0.

Q2

 

Find the equation of the circle which touches the circle

x2 + y2 – 6x + 6y + 17 = 0 externally and to which the lines

x2 – 3xy – 3x + 9y = 0 are normals.

Q3

 

Find the equation of a circle which passes through the point

(2, 0) and whose centre is the limit of the point of intersection of the lines 3x + 5y = 1and (2 + c)x + 5c2y = 1as c → 1.

Q4

Tangents are drawn from (6, 8) to the circle x2 + y2 = r2. Find the radius of the circle such that the areas of the âˆ† formed by tangents and chord of contact is maximum.

Q5

Find the radius of smaller circle which touches the straight line 3x – y = 6 at (1, –3) and also touches the line y = x.

Q6

2x – y + 4 = 0 is a diameter of the circle which circumscribed a rectangleABCD. If the co-ordinates of A and B are A(4, 6) and B(1, 9), find the area of rectangle ABCD.

Q7

Let 2x2 + y2 – 3xy = 0 be the equation of a pair of tangents drawn from the origin O to a circle of radius 3 with centre in the first quadrant. If A is one of the points of contact, find the length of OA.

Q8

If the circle C1x2 + y2 = 16 intersects another circle C2 of radius 5 in such a manner that the common chord is of maximum length and has a slope equal to (3/4), find the co-ordinates of centre C2.

Q9

The circle x2 + y2 = 1 cuts the x-axis at P and Q. another circle with centre at Q and variable radius intersects the first circle at R above the x-axis and the line segment PQ at S. Find the maximum area of the triangleQSR.

Q10

 

Find the equation of the circle whose radius is 5 and which touches the circle 

              x2 + y2 – 2x – 4y – 20 = 0 at the point (5, 5).