Question

Passage: P(2t², 4t) is a point on the parabola y² = 8x and Q(h, k) is a point on the tangent at P and external to the circle x² + y² = 8. 

Answer the following questions.

(i) As Q moves on the tangent at P, the locus of the point of intersection of the chord of contact of Q with respect to the circle at the tangent at P is

(A)  y² - x² = 4

(B)  y² = 2x² = 4

(C)  y² - 2x² = 4

(D)  y² = -4x

(ii) The point in the second quadrant from which perpendicular tangents can be drawn to both the parabola and the circle is

(A) (-2,2√3)

 (B)  (-√2, √2)

(C)  (-√2,√2)

(D)  (-3,√3)

(iii)  If AB is the chord of contact of Q(h, k) with respect to the circle x² + y² = 8, then the circumcentre of ΔAQB lies on the curve (when t = 2)

(A)   x + 2y = 4

(B)   x -  2y = 4

(C)  2y = x + 4

(D)   x + 2y + 4 = 0

Answer 1

Correct option  (i)(D)(ii)(A)(iii)(C)

Explanation :

(i) The tangent at is

ty = x + 2t² ..(i)

which passes through Q(h, k). This implies

tk = h + 2t² ..(ii)

The equation of the chord of contact of Q(h, k) with respect to the circle x² + y² = 8 .....(iii)

From Eqs. (2) and (3), we have

 This line passes through the point

 y = 4/t, x = -y/t

 , Therefore, the locus is

y/4 = -x/y

 y² = -4x

(ii) The required point lies on the director circle of the circle and directrix of the parabola. The directrix of the parabola is x + 2 = 0 and the director circle of the given circle is 

x = -2 ⇒ 4 + y² = 16

⇒ y = ±2√3

 2 Therefore, the required point = (-2,2√3).

(iii) The equation of the circumcircle of ΔAQB

(x² + y²  - 8) + λ(hx + ky - 8) = 0

This should pass through (0, 0) (centre of the circle) which implies that  λ = - 1. Therefore, the circumcircle of ΔAQB is

x² + y² - hx - ky = 0

so that (h/2,k/2) is its centre. If (x, y) is the circumcentre of ΔAQB, then x = h/2,y = k/2. . Substituting the values of h and k in Eq. (2), when t = 2,

we have

2(2y) = 2x + 8

⇒ x - 2y + 4 = 0