Question

P is a variable on the line y = 4. Tangents are drawn to the circle x2 + y2= 4 from P to touch it at A and B. The perpendicular PAQB is completed. Find the equation of the locus of Q.

Solution

Correct option is

(x2 + y2)(y + 4) = 2y

 

Let P (h, 4) be a variable point. Given circle is 

                x2 + y2 = 4                 …(1)

Draw tangents from P (h, 4) and complete parallelogram PAQB.

Equation of the diagonal AB which is chord of contact of

x2 + y= 4 is  

             hx + 4y = 4              …(2)

Let co-ordinates of A and B are (x1y1) and (x2y2) respectively.

Since (x1y1) and B (x2y2) lies on (2)  

∴           hx1 + 4y1 = 4   

and       hx2 + 4y2 = 4  

∴          h(x1 + x2) + 4(y1 + y2) = 8                  …(3)

Since PAQB is parallelogram  

∴         mid point of AB = mid point of PQ

  

Eliminating x from (1) and (2) then

 

   

  

  

From (3) and (5), we get        

From (4) and (6)                     

from (4) and (6) 

                            

or                         (16 + h2)(α + h) = 8h                       …(8) 

Dividing (8) by (7) then        

Substituting the value h in (7) then

                           

  

Hence locus of Q (α, β) is (x2 + y2)(y + 4) = 2y2

SIMILAR QUESTIONS

Q1

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.

Q2

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.

Q3

 

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. 

Q4

 

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).   

Q5

Find the locus of the mid point of the chord of the circle x2 + y2 = a2which subtend a right angle at the point (pq).

Q6

 

Let a circle be given by

                   2x (x – a) + y(2y – b) = 0            (a ≠ 0, b ≠ 0)

Find the condition on a and b if two chords, each bisected by the x-axis, can be drawn to the circle from (ab/2).

Q7

The centre of the circle S = 0 lie on the line 2x – 2y + 9 = 0 and S = 0 cuts orthogonally the circle x2 + y2 = 4. Show that circle S = 0 passes through two fixed points and find their co-ordinates.

Q8

 be a given circle. Find the locus of the foot of perpendicular drawn from origin upon any chord of Swhich subtends a right angle at the origin.

Q9

 be a given circle. Find the locus of the foot of perpendicular drawn from origin upon any chord of Swhich subtends a right angle at the origin.

Q10

 

Find the condition on abc such that two chords of the circle

                x2 + y2 – 2ax – 2by + a2 + b2 – c2 = 0  

passing through the point (ab + c) are bisected by the line y = x.