Conic Sections

Let C be the circle x² + (y - 1)² = 2, E? and E? be two ellipses whose centres lie at the origin and major axes lie on x-axis and y-axis respectively. Let the straight line x + y = 3 touch the curves C, E?, and E? at P(x?, y?), Q(x?, y?), and R(x?, y?) respectively. Given that P is the mid-point of the line segment QR and PQ = 2√2/3, the value of 9(x?y? + x?y? + x?y?) is equal to ___.

The radius of the smallest circle which touches the parabolas y = x² + 2 and x = y² + 2 is

Let e₁ and e₂ be the eccentricities of the ellipse, x²/25 + y²/b² = 1 (b < 5) and the hyperbola, x/16 - y/b = 1 respectively satisfying ee = 1. If and are the distances between the foci of the ellipse and the foci of the hyperbola respectively, then the ordered pair (, ) is equal to :

Let the latus rectum of the parabola y² = 4x be the common chord to the circles C₁ and C₂ each of them having radius 2√5. Then, the distance between the centres of the circles C₁ and C₂ is:

The diameter of the circle, whose centre lies on the lines x + y = 2 in the first quadrant and which touches both the lines x = 3 and y = 2, is

Let P be a point on the parabola, y² = 12x and N be the foot of the perpendicular drawn from P on the axis of the parabola. A line is now drawn through the mid-point M of PN, parallel to its axis which meets the parabola at Q. If the y-intercept of the line NQ is 4/3, then:

A hyperbola having the transverse axis of length √2 has the same foci as that of the ellipse 3x² + 4y² = 12, then this hyperbola does not pass through which of the following points?

Let the point P of the focal chord PQ of the parabola y² = 16x be (1, −4). If the focus of the parabola divides the chord PQ in the ratio m: n, gcd(m, n) = 1, then m² + n² is equal to :

The area (in sq. units) of an equilateral triangle inscribed in the parabola y²=8x, with one of its vertices on the vertex of this parabola, is