Question

Tangents PQPR are drawn to the circle x2 + y2 = 36 from the point P(–8, 2) touching the circle at QR respectively. Find the equation of the circumcircle of the âˆ†PQR.

Solution

Correct option is

x2 + y2 + 8x – 2y = 0

 

Equation of the required circle from point (7) is

              (x + 8) (x – 0) + (y – 2) (y – 0) = 0  

or           x2 + y2 + 8x – 2y = 0 

Alternative Method: Equation of the chord of contact of the circle x2 +y2 = 36 with respect to the point P(–8, 2) is

                    –8x + 2y = 36

or            4x – y + 18 = 0               (Equation of AB)

Equation of the required circle through point of intersection of x2 + y2 = 36 and 4x – y + 18 = 0   

is             (x2 + y2 – 36) + λ(4x – y + 18) = 0                …(1)

which passes through P(–8, 2) then 

                (64 + 4 – 36) + λ(–32 – 2 + 18) = 0  

or             λ = 2

Substituting the value of λ = 2 in (1), then the required circle is

            x2 + y2 + 8x – 2y = 0 

SIMILAR QUESTIONS

Q1

 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.

Q2

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.

Q3

 

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.  

Q4

 

Find the limiting points of the circles 

       (x2 + y2 + 2gx + c) + λ(x2 + y2 + 2fy + d) = 0

Q5

The circle x2 + y2 – 4x – 8y + 16 = 0 rolls up the tangent to it at  by 2 units, assuming the x-axis as horizontal, find the equation of the circle in the new position.  

Q6

 

Find the equation of the circle of minimum radius which contains the three circles 

                   x2 – y2 – 4y – 5 = 0 

               x2 + y2 + 12x + 4y + 31 = 0  

and         x2 + y2 + 6x + 12y + 36 = 0

Q7

Find the equation of the circle passing through (1, 0) and (0, 1) and having the smallest possible radius.

Q8

 

If the line x cos α + y sin α = p cuts the circle x2 + y2 = a2 in M and N, then show that the circle, whose diameter is MN, is 

              

Q9

A line meets the co-ordinate axes at A and B. A circle is circumscribed about the triangle OAB. If the distance of the points A and B from the tangent at O, the origin, to the circle are m and n respectively, find the equation of the circle.

Q10

Let 1 and C2 be two circles with C2 lying inside C1. A circle C lying inside C1 touches C1 internally and C2 externally. Identify the locus of the centre of C.