Question

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.

Solution

Correct option is

 

Let the centre of the circle C2 is Q(hk), Equation of the circle C2 is

                    

or          x2 + y2 – 2xh – 2yk + h2 + k2 – 25 = 0  

              

And equation of circle   

∴ Equation of common chord is

                       C1 – C2 = 0  

  

or        2hx + 2ky – (h2 + k2 – 9) = 0                   …(1)

           Slope of this line = –h/k

But, it is given that its slope = 3/4

  

or                       3k + 4h = 0                        …(2)

Let P be the length perpendicular from P(0, 0) on chord (1), then

                              

Length of this chord AB = 2AM  

                                         

This chord has maximum length, then p = 0 then from (3), 

                             h2 + k2 – 9 = 0                        …(4)

Solving (2) and (4) we get 

                              

  

SIMILAR QUESTIONS

Q1

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.

Q2

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.

Q3

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

Q4

 

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.

Q5

 

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.

Q6

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.

Q7

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

Q8

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.

Q9

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.

Q10

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.