Conic Sections

Let a circle C touch the lines L₁ : 4x – 3y + K₁ = 0 and L₂ : 4x – 3y + K₂ = 0, K₁, K₂  $\in$  R. If a line passing through the centre of the circle C intersects L₁ at (-1,2) and L₂ at (3, -6), then the equation of the circle C is:

The locus of the mid- point of the line segment joining the focus of the parabola y² = 4ax to a moving point of the parabola, is another parabola whose directrix is:

A circle passes through the points (2, 2) and (9, 9) and touches the x-axis. The absolute value of the difference of possible x-coordinate of the point of contact is __________

The value of 2 cos 20º cos 40° cos 80° is equal to __________

The eccentricity of the conic represented by the parametric equation x = p - t + 1, y = p + t + 1

Let E be an ellipse whose axes are parallel to the co-ordinates axes, having its center at (3,-4), one focus at (4,-4) and one vertex at (5,-4). If mx - y = 4, m > 0 is a tangent to the ellipse E, then the value of 5m² is equal to……….

Consider a circle C which touches the y-axis at (0, 6) and cuts off an intercept 6√5 on the x-axis. Then the radius of the circle C is equal to :

A circle of radius 2 unit passes through the vertex and the focus of the parabola y² = 2x and touches the parabola y = 2x and touches the parabola y = (x-1/4)² + a, where a > 0. Then (4α - 8)² is equal to.

A rectangle R with end points of one of its sides as (1, 2) and (3, 6) is inscribed in a circle. If the equation of a diameter of the circle is 2x - y + 4 = 0, then the area of R is.

Let the eccentricity of an ellipse x²/a² + y²/b² = 1, a>b, be 1/4. If this ellipse passes through the point (-4√(2/5), 3), then a² + b² is equal to: