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.
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.
(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