Conic Sections

Let x = 4 be a directrix to an ellipse whose centre is at the origin and its eccentricity is 1/2. If P(1, β), β > 0 is a point on this ellipse, then the equation of the normal to it at P is:

The circle passing through the intersection of the circles, x² + y² – 6x = 0 and x² + y² - 4y = 0, having its centre on the line, 2x – 3y + 12 = 0, also passes through the point:

Let P(3,3) be a point on the hyperbola, x²/a² - y²/b² = 1. If the normal to it at P intersects the x-axis at (9,0) and e is its eccentricity, then the ordered pair (a², e²) is equal to:

Let x²/a² + y²/b² = 1(a > b) be given ellipse, length of whose latus rectum is 10. If its eccentricity is the maximum value of the function, φ(t) = 5/12 + t − t², then a² + b² is equal to:

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 ___.

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