Analytic Geometry

Question 1

Given the equation of the ellipse: 25x² + 9y² – 250x – 54y + 481 = 0. Compute the distance between foci.

Answer 1

4

Question 2

Find the distance between P1 ( 3, 120°) and P2 ( 4, 30°).

Answer 2

5

Question 3

A circle has its center at (4, -2) and one end of its diameter is at (5, 3). Find the other end of this diameter.

Answer 3

(3, -7)

Question 4

Determine the point of division nearest to A that divides the line segment from A( 7,-2 ) to B(-2, 7 ) into 2 parts in the ratio 4:5.

Answer 4

(3, 2)

Question 5

Compute the eccentricity of the curve r = 2/(1+ sin ϴ).

Answer 5

1

Question 6

Find the equation of the parabola with focus at (1, 1) and x = 7 as directrix.

Answer 6

y² – 2y + 12x – 11 = 0

Question 7

The angle from the line 3x – 7y + 6 = 0 to the line 2x + By + 12 = 0 is 45°. Find B.

Answer 7

1

Question 8

Find the equation of the parabola whose axis is parallel to the x-axis and which passes through the three points (0, 0), (8, -4) and (3, 1).

Answer 8

1

Question 9

Find the distance from the point (5, - 3) to the line 7x – 4y – 28 = 0.

Answer 9

1

Question 10

Find the area of the circle with center at (2, - 4) and tangent to the circle whose equation is x² + y² – 4x – 4y + 4 = 0.

Answer 10

1

Question 11

The vertices of an equilateral triangle are A( - 1, 4 ), B( x, - 2 ) and C( 3 √2 , 1). Find x.

Answer 11

-1

Question 12

Find the distance between the parallel lines 3x – 4y + 4 = 0 and 3x – 4y – 16 = 0.

Answer 12

1

Question 13

Find the center of the hyperbola 16x² – 9y² – 32x – 36y – 164 = 0.

Answer 13

1

Question 14

The ceiling in a hallway 10m wide is in the shape of a semi - ellipse and is 9m high at the center and 6m high at the sidewalls. Find the height of the ceiling 3m from either wall.

Answer 14

1

Question 15

Find the equation of the line with x-intercept 5 and y-intercept of -3.

Answer 15

1

Question 16

An elliptical field has major axis 30 m and minor axis 24 m. If the cost of fencing the field is P 1,500 per meter length, what is the total cost of enclosing the perimeter of the field?

Answer 16

1

Question 17

Find the equation of the circle whose diameter connects and run through the points ( 3, - 3 ) and ( 9, 5 ) in the Cartesian coordinate plane.

Answer 17

1

Question 18

An ellipse has its major axis on the x-axis, center at ( -3, 0 ), one of the vertices at ( -5, 0 ) and length of latus rectum equal to 1. Find its equation.

Answer 18

1

Question 19

Find y if A( 9, y ), B( 4, 3 ) and C( - 3, - 7 ) are vertices of a right triangle with the right angle at B.

Answer 19

1

Question 20

When the load is uniformly distributed horizontally, the cable of a suspension bridge hangs in the form of a parabola. The supporting towers of a suspension bridge are 40 m high and spaced 60 m apart. The lowest point of the cable is 12 m above the roadway. Find the vertical distance from the roadway to the cable at a point on the roadway 10 m from either tower.

Answer 20

1

Question 21

Find the area of a polygon with vertices at ( -5, 0 ), ( 2, - 4 ), ( 5, 2 ), ( 3, 5 ) and ( 0, 3 ).

Answer 21

1

Question 22

Find the equation of the line through ( 2, 8 ) and perpendicular to x – 2y + 8 = 0.

Answer 22

1

Question 23

Find the equation of the hyperbola with center at the origin, a vertex at ( 0, 3 ) and a focus at ( 0, 4 ).

Answer 23

1

Question 24

Given the equation: x2 – 10x + 4y = 196 – y2. Find the shortest distance from the curve to the point ( 20, 25 ).

Answer 24

1

Question 25

Given the equation of the parabola y² + 16y + 8x + 40 = 0. Find the coordinates of its vertex.

Answer 25

1

Question 26

Find the equation of the line with normal angle 45° and with normal intercept.

Answer 26

1

Question 27

Find the equation of the hyperbola with center at the origin, a vertex at ( 0, 3 ) and a focus at ( 0, 4 ).

Answer 27

1

Question 28

A conic section is described by the equation rcos 2θ = sinθ. Compute the length of the latus rectum.

Answer 28

1

Question 29

The parametric equation of a curve is given as x = 4t – 3, y = 16t² – 9. Compute the length of the latus rectum.

Answer 29

1

Question 30

Transform the parametric equations x = cos θ , y = cos² θ + 8 cos θ into its corresponding Cartesian equation.

Answer 30

1