Question

Find the equations of the tangents from the point A(3, 2) to the circle x2y2 + 4x + 6y + 8 = 0 .

Solution

Correct option is

2x – y – 4 = 0 and x – 2y + 1 = 0

 

Combined equation of the pair of tangents drawn from A(3, 2) to the given circle x2 + y2 + 4x + 6y + 8 = 0 can be written in the usual notation.

                     

   

            = [x2 + y2 + 4x + 6y + 8] [9 + 4 + 12 + 12 + 8]

⇒            (5x + 5y + 20)2 = 45(x2 + y2 + 4x + 6y + 8)

⇒            5(x + y + 4)2 = 9(x2 + y2 + 4x + 6y + 8)

⇒         5(x2 + y2 + 2xy + 8x + 8y + 16) = 9(x2 + y2 + 4x + 6y + 8)

⇒         4x2 + 4y2 – 10xy – 4x + 14y – 8 = 0

or         2x2 + 2y2 – 5xy – 2x + 7y – 4 = 0

or         (2x – y – 4) (x – 2y + 1) = 0  

hence the required tangents to the circle from A(3, 2) are

            2x – y – 4 = 0 and x – 2y + 1 = 0

SIMILAR QUESTIONS

Q1

Find the area of the triangle formed by tangents from the point (4, 3) to the circle x2 + y2 = 9 and the line segment joining their points of contact is 

Q2

Find the length of the tangent from any point on the circle x2 + y2 + 2gx+ 2fy + c = 0 to the circle x2 + y2 + 2gx + 2fy + c1 = 0 is

Q3

 

Find the power of point (2, 4) with respect to the circle 

                x2 + y2 – 6x + 4y – 8 = 0

Q4

Show that the locus of the point, the powers of which with respect to two given circles are equal, is a straight line.

Q5

 

Find the condition that chord of contact of any external point

(hk) to the circle x2 + y2 = a2 should subtend right angle at the centre of the circle.

Q6

The chord of contact of tangents drawn from a point on the circle x2 +y2 = a2 to the circle x2 + y2 = b2 touches the circle x2 = y2 = c2. Show that abc are in GP.

Q7

Find the equation of the chord x2 + y2 – 6x + 10y – 9 = 0 which is bisected at (–2, 4).

Q8

Find the middle point of the chord intercepted on line lx + my + n = 0 by the circle x2 + y2 = a2.

Q9

Find the locus of middle points of chords of the circle x2 + y2 = a2, which subtend right angle at the point (c, 0).

Q10

If two tangents are drawn from a point on the circle x2 + y2 = 50 to the circle x2 + y2 = 25 then find the angle between the tangents.