Question

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

Solution

Correct option is

1.5

 

Let C (hk) be the centre of the circle. Let AB and C’D be the lines represented by 3x – y = 6 and y = x respectively.

Clearly, the circle touches AB at A (1, –3).

Equation of line ⊥ to 3x – y = 6 is x + 3y = λ which passes through (1, –3).

Then           1 – 9 = λ    

    

  

Which passes through C (hk)   

then            h + 3k + 8 = 0                     …(1)

Now centre     

                   

  

  

  

  

Since radius from (2),  

            

  

                                     

  

                                    

Hence radius of smaller circle is 1.5 units.

SIMILAR QUESTIONS

Q1

Find the lengths of common tangents of the circles x2 + y2 = 6x and x2 +y2 + 2x = 0.  

Q2

 

Find the equation of the circle circumscribing the triangle formed by the lines:

         x + y = 6, 2x + y = 4 and x + 2y = 5,

Without finding the vertices of the triangle.

Q3

Find the equation of the circle circumscribing the quadrilateral formed by the lines in order are 5x + 3y – 9 = 0, x – 3y = 0, 2x – y = 0, x + 4y – 2 = 0 without finding the vertices of quadrilateral.

Q4

Find the equation of a circle which touches the x-axis and the line 4x – 3y+ 4 = 0. Its centre lies in the third quadrant and lies on the line x – y – 1 = 0.

Q5

Find the equations of the circle which passes through the origin and cut off chords of length a from each of the lines y = x and y = –x.

Q6

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

Q7

 

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.

Q8

 

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.

Q9

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.

Q10

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.