Chord of Contact

From a point P(x1, y1) two tangents PA and PB can be drawn to the circle. The chord AB joining the points of contact A and B of the tangents from P is called the chord of contact of P(x1, y1) with respect to the circle. Its equation is given by T = 0.

Example;  Let P be any moving point on the circle x2 + y2 − 2x = 1 , from this point chord of contact is drawn w.r.t. the circle x2 + y2 − 2x = 0. Find the locus of the circumcentre of the triangle CAB, C being centre of the circle.

Solution:

The two circles are

(x − 1)2 + y2 = 1 … (i)

(x − 1)2 + y2 = 2 ….. (ii)

So the second circle is the director circle of the first. So ∠APB = π/2.

Also ∠ACB = π/2.
Now circumcentre of the right angled triangle CAB would lie on the mid-point of AB.

So, let the point be M ≡ (h, k)

Now, CM = CB sin45° = 1/√2

So, (h −1)2 + k2 = (1/√2)2

So, locus of M is (x − 1)2 + y2 = 1/2

Example :. Find the value of λ so that the line 3x − 4y = λ may touch the circle x2 + y2 − 4x − 8y − 5 = 0

Solution: Solving both curves, we get

x2 + (3x − λ)2/42 − 4x − 8(3x − λ)/4 − 5 = 0

=> 25x2 − (6 + 160)x + λ2 + 32λ − 80 = 0

as given line touches the circle

=> D = 0

=> (6λ + 160)2 − 4 x 25(λ2 + 32λ − 80) = 0

=> λ2 + 20λ − 525 = 0

=> (λ+ 35)(λ − 15) = 0

=> λ = −35 , 15.

Exercise :

(i) Prove that the tangent to the circle x2 + y2 = 5 at the point (1, −2) also touches the circle x2 + y2 − 8x +6y + 20 = 0 and find its point of contact.

(ii) From a point P tangents drawn to the circles x2+ y2 + x − 3 = 0, 3x2 + 3y2 − 5x + 3y = 0 and 4x2 + 4y2 + 8x + 7y + 9 = 0 are of equal lengths. Find the equation of the circle through P which touches the line x + y = 5 at the point (6, −1)

(iii) If from any point on the circle x2 + y2 + 2gx + 2fy + c = 0 tangents are drawn to the circle x2 + y2 + 2gx + 2fy + c sin2α + (g2 + f2) cos2α = 0, show that the angle between the tangents is equal to 2α.

(iv) The extremities of a diagonal of a rectangle are (−4, 4) and (6, −1). A circle circumscribe the rectangle and cuts an intercept AB on the y-axis. Find the area of the triangle formed by AB and the tangents to the circle at A and B.

(v) Find the equations to the circles in which the line joining the points (a , b) and (b , −a) is a chord subtending an angle of 45° at any point on its circumference.

Also Read :

Equations of Circle in various forms
Parametric Equation of a Circle
Position of a Point w.r.t a Circle
Equation of Tangent & Normal on the Circle
Radical Axis of two circles
Equation of Family of Circles
External & Internal Contacts of Circles

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