PRACTICE PROBLEMS ON PARABOLA ELLIPSE AND HYPERBOLA IN CONIC SECTIONS

Problem 1 :

The eccentricity of the hyperbola whose latus rectum is 8 and conjugate axis is equal to half of the distance between the foci is 

a)  4/3      b) 4/√3      c)  2/√3        d)  3/2

Solution

Problem 2 :

If P(x, y) be any point on 16x² + 25y² = 400 and foci F1(-3, 0) then PF1 + PF2 is 

a)  8     b)  6    c)  10     d)  12

Solution

Problem 3 :

The area of quadrilateral formed with foci of the hyperbolas 

a)  4(a² + b²)      b) 2(a² + b²)

c) (a² + b²)           d) 1/2(a² + b²)

Solution

Problem 4 :

If the normals of the parabola y² = 4x drawn at the end points of its latus rectum are tangents to the circle (x - 3)² + (y + 2)² = r², then the value of r² is 

a)  2      b)  3      c)   1       d)      4

Solution

Problem 5 :

If x+y = k is a normal to the parabola y² = 12x, then the value of k is 

a)  3    b)  -1    c)  1      d)  9

Solution

Problem 6 :

The ellipse E1 :

is inscribed in a rectangle R whose sides are parallel to the coordinate axes. Another ellipse E2 passing through the point (0, 4) circumscribing the rectangle R. The eccentricity of the ellipse is 

a) √2/2    b)  √3/2    c)  1/2      d)  3/4

Solution

Problem 7 :

Tangents are drawn to the hyperbola 

parallel to he straight line 2x - y = 1. One of the points of contact of tangents on the hyperbola is 

Solution

Problem 8 :

The equation of the circle passing through the foci of the ellipse 

having center at (0, 3) is

a)  x² + y² - 6y - 7 = 0         b)  x² + y² - 6y + 7 = 0

c)  x² + y² - 6y - 5 = 0         d)  x² + y² - 6y + 5 = 0

Solution

Problem 9 :

Consider an ellipse whose center is of the origin and its major axis is along x-axis. If its eccentricity is 3/5 and the distance between its foci is 6, then the area of the quadrilateral in the ellipse with diagonals as major and minor axis of the ellipse is 

a)  8         b)  32    c) 80         d)  40

Solution

Problem 10 :

Area of the greatest rectangle inscribed in the ellipse 

a) 2ab       b) ab     c)  √ab       d)  a/b

Solution

Problem 11 :

An ellipse has OB as semi minor axes, F and F' its foci and the angle FBF' is a right angle then the eccentricity of ellipse is 

a) 1/√2       b) 1/2     c)  1/4       d)  1/√3

Solution

Problem 12 :

The eccentricity of the ellipse (x - 3) + (y - 4)² = y²/9

a) √3/2       b) 1/3     c)  1/3√2      d)  1/√3

Solution

Problem 13 :

If the two tangents from a point P to the parabola y² = 4x are at right angles then locus P is 

a) 2x + 1 = 0       b) x = -1     c)  2x - 1 = 0      d)  x = 1

Solution

Problem 14 :

The locus of a point whose distance from (-2, 0) is 2/3 times its distance from the line x = -9/2 is

a) a parabola       b) hyperbola     c)  ellipse      d)  circle

Solution

Problem 15 :

The value of m for which the line y = mx + 2√5 touches the hyperbola 16x² - 9y² = 144 are the roots of x² - (a + b)x - 4 = 0, then the value of (a + b) is

a) 2       b) 4   c)  9      d)  -2

Solution