This is the Multiple Choice Questions Part 1 of the Series in Analytic Geometry: Parabola, Ellipse and Hyperbola topics in Engineering Mathematics. In Preparation for the ECE Board Exam make sure to expose yourself and familiarize in each and every questions compiled here taken from various sources including but not limited to past Board Examination Questions in Engineering Mathematics, Mathematics Books, Journals and other Mathematics References.

#### MCQ Topic Outline included in Mathematics Board Exam Syllabi

- MCQ in Rectangular coordinates system | MCQ in Distance Formula | MCQ in Distance between two points in space | MCQ in Slope of a Line | MCQ in Angle between two lines | MCQ in Distance between a point and a line | MCQ in Distance between two lines | MCQ in Division of line segment | MCQ in Area by coordinates | MCQ in Lines | MCQ in Conic sections | MCQ in Circles

#### Start Practice Exam Test Questions Part 1 of the Series

**Choose the letter of the best answer in each questions.**

**Problem 1: CE Board May 1995**

What is the radius of the circle x^{2} + y^{2} โ 6y = 0?

A. 2

B. 3

C. 4

D.5

Answer: **Option B**

Solution: Review: Solution for Number 1

**Problem 2: CE Board November 1995**

What are the coordinates of the center of the curve x^{2} + y^{2} โ 2x โ 4y โ 31 = 0?

A. (-1, -1)

B. (-2, -2)

C. (1, 2)

D. (2, 1)

Answer: **Option C**

Solution: Review: Solution for Number 2

**Problem 3:**

A circle whose equation is x^{2} + y^{2} + 4x +6y โ 23 = 0 has its center at

A. (2, 3)

B. (3, 2)

C. (-3, 2)

D. (-2, -3)

Answer: **Option D**

Solution: Review: Solution for Number 3

**Problem 4: ME Board April 1998**

What is the radius of a circle with the ff. equation: x^{2} โ 6x + y^{2} โ 4y โ 12 = 0

A. 3.46

B. 7

C. 5

D.6

Answer: **Option C**

Solution: Review: Solution for Number 4

**Problem 5: ECE Board April 1998**

The diameter of a circle described by 9x^{2} + 9y^{2} = 16 is?

A. 4/3

B. 16/9

C. 8/3

D. 4

Answer: **Option C**

Solution: Review: Solution for Number 5

**Problem 6: CE Board May 1996**

How far from the y-axis is the center of the curve 2x^{2} + 2y^{2} +10x โ 6y โ 55 = 0

A. -2.5

B. -3.0

C. -2.75

D. -3.25

Answer: **Option A**

Solution: Review: Solution for Number 6

**Problem 7: **

What is the distance between the centers of the circles x^{2} + y^{2} + 2x + 4y โ 3 = 0 and x^{2} + y^{2} โ 8x โ 6y + 7 = 0?

A. 7.07

B. 7.77

C. 8.07

D. 7.87

Answer: **Option A**

Solution: Review: Solution for Number 7

**Problem 8: CE Board November 1993**

The shortest distance from A (3, 8) to the circle x^{2} + y^{2} + 4x โ 6y = 12 is equal to?

A. 2.1

B. 2.3

C. 2.5

D. 2.7

Answer: **Option A**

Solution: Review: Solution for Number 8

**Problem 9: ME Board October 1996**

The equation circle x^{2} + y^{2} โ 4x + 2y โ 20 = 0 describes:

A. A circle of radius 5 centered at the origin.

B. An eclipse centered at (2, -1).

C. A sphere centered at the origin.

D. A circle of radius 5 centered at (2, -1).

Answer: **Option D**

Solution: Review: Solution for Number 9

**Problem 10: EE Board April 1997**

The center of a circle is at (1, 1) and one point on its circumference is (-1, -3). Find the other end of the diameter through (-1, -3).

A. (2, 4)

B. (3, 5)

C. (3, 6)

D. (1, 3)

Answer: **Option B**

Solution: Review: Solution for Number 10

**Problem 11:**

Find the area (in square units) of the circle whose equation is x^{2} + y^{2} = 6x โ 8y.

A. 20 ฯ

B. 22 ฯ

C. 25 ฯ

D. 27 ฯ

Answer: **Option C**

Solution: Review: Solution for Number 11

**Problem 12:**

Determine the equation of the circle whose radius is 5, center on the line x = 2 and tangent to the line 3x โ 4y + 11 = 0.

A. (x โ 2)^{2} + (y โ 2)^{2} = 5

B. (x โ 2)^{2} + (y + 2)^{2} = 25

C. (x โ 2)^{2} + (y + 2)^{2} = 5

D. (x โ 2)^{2} + (y โ 2)^{2} = 25

Answer: **Option B**

Solution: Review: Solution for Number 12

**Problem 13:**

Find the equation of the circle with the center at (-4, -5) and tangent to the line 2x + 7y โ 10 = 0.

A. x^{2} + y^{2} + 8x โ 10y โ 12 = 0

B. x^{2} + y^{2} + 8x โ 10y + 12 = 0

C. x^{2} + y^{2} + 8x + 10y โ 12 = 0

D. x^{2} + y^{2} โ 8x + 10y + 12 = 0

Answer: **Option C**

Solution: Review: Solution for Number 13

**Problem 14: ECE Board April 1998**

Find the value of k for which the equation x^{2} + y^{2} + 4x โ 2y โ k = 0 represents a point circle.

A. 5

B. 6

C. -6

D. -5

Answer: **Option D**

Solution: Review: Solution for Number 14

**Problem 15: ECE Board April 1999**

3x^{2} + 2x โ 5y + 7 = 0. Determine the curve.

A. Parabola

B. Ellipse

C. Circle

D. Hyperbola

Answer: **Option A**

Solution: Review: Solution for Number 15

**Problem 16: CE Board May 1993, CE Board November 1993, ECE Board April 1994**

The focus of the parabola y^{2} = 16x is at

A. (4, 0)

B. (0, 4)

C. (3, 0)

D. (0, 3)

Answer: **Option A**

Solution: Review: Solution for Number 16

**Problem 17: CE Board November 1994**

Where is the vertex of the parabola x^{2} = 4(y โ 2)?

A. (2, 0)

B. (0, 2)

C. (3, 0)

D. (0, 3)

Answer: **Option B**

Solution: Review: Solution for Number 17

**Problem 18: ECE Board April 1994, ECE Board April 1999**

Find the equation of the directrix of the parabola y^{2} = 16x.

A. x = 2

B. x = -2

C. x = 4

D. x = -4

Answer: **Option D**

Solution: Review: Solution for Number 18

**Problem 19:**

Given the equation of a parabola 3x + 2y^{2} โ 4y + 7 = 0. Locate its vertex.

A. (5/3, 1)

B. (5/3, -1)

C. -(5/3, -1)

D. (-5/3, 1)

Answer: **Option D**

Solution: Review: Solution for Number 19

**Problem 20: ME Board April 1997**

In the equation y = – x^{2} + x + 1, where is the curve facing?

A. Upward

B. Facing left

C. Facing right

D. Downward

Answer: **Option D**

Solution: Review: Solution for Number 20

**Problem 21: CE Board May 1995**

What is the length of the length of the latus rectum of the curve x^{2} = 20y?

Answer: **Option B**

Solution: Review: Solution for Number 21

**Problem 22: EE Board October 1997**

Find the location of the focus of the parabola y^{2} + 4x โ 4y โ 8 = 0.

A. (2.5, -2)

B. (3, 1)

C. (2, 2)

D. (-2.5, -2)

Answer: **Option C**

Solution: Review: Solution for Number 22

**Problem 23: ECE Board April 1998**

Find the equation of the axis of symmetry of the function y = 2x^{2} โ 7x + 5.

A. 7x + 4 = 0

B. 4x + 7 = 0

C. 4x โ 7 = 0

D. x โ 2 = 0

Answer: **Option C**

Solution: Review: Solution for Number 23

**Problem 24:**

A parabola has its focus at (7, -4) and directrix y = 2. Find its equation.

A. x^{2} + 12y โ 14x + 61 = 0

B. x^{2} โ 14y + 12x + 61 = 0

C. x^{2} โ 12x + 14y + 61 = 0

D. none of these

Answer: **Option A**

Solution: Review: Solution for Number 24

**Problem 25:**

A parabola has its axis parallel to the x-axis, vertex at (-1, 7) and one end of the latus rectum at (-15/4, 3/2). Find its equation.

A. y^{2} โ 11y + 11x โ 60 = 0

B. y^{2} โ 11y + 14x โ 60 = 0

C. y^{2} โ 14y + 11x + 60 = 0

D. none of these

Answer: **Option C**

Solution: Review: Solution for Number 25

**Problem 26: ECE Board November 1997**

Compute the focal length and the length of the latus rectum of the parabola y^{2} + 8x โ 6y + 25 = 0.

A. 2, 8

B. 4, 16

C. 16, 64

D. 1, 4

Answer: **Option A**

Solution: Review: Solution for Number 26

**Problem 27:**

Given a parabola (y โ 2)^{2} = 8(x โ 1). What is the equation of its directrix?

A. x = -3

B. x = 3

C. y = -3

D. y = 3

Answer: **Option B**

Solution: Review: Solution for Number 27

**Problem 28: ME Board October 1997**

The general equation of a conic section is given by the following equation: Ax^{2} + Bxy + Cy^{2} + Dx + Ey + F = 0. A curve maybe identified as an ellipse by which of the following conditions?

A. B^{2} โ 4AC < 0

B. B^{2} โ 4AC = 0

C. B^{2} โ 4AC > 0

D. B^{2} โ 4AC = 1

Answer: **Option A**

Solution: Review: Solution for Number 28

**Problem 29: CE Board November 1994**

What is the area enclosed by the curve 9x^{2} + 25y^{2} โ 225 = 0?

A. 47.1

B. 50.2

C. 63.8

D. 72.3

Answer: **Option A**

Solution: Review: Solution for Number 29

**Problem 30: ECE Board April 1998**

Point P (x, y) moves with a distance from point (0, 1) one-half of its distance from line y = 4. The equation of its locus is?

A. 2x^{2} โ 4y^{2} = 5

B. 4x^{2} + 3y^{2} = 12

C. 2x^{2} + 5y^{3} = 3

D. x^{2} + 2y^{2} = 4

Answer: **Option B**

Solution: Review: Solution for Number 30

**Problem 31:**

The lengths of the major and minor axes of an ellipse are 10 m and 8 m, respectively. Find the distance between the foci.

A. 3

B. 4

C. 5

D. 6

Answer: **Option D**

Solution: Review: Solution for Number 31

**Problem 32:**

The equation 25x^{2} + 16y^{2} โ 150x + 128y + 81 = 0 has its center at?

A. (3, -4)

B. (3, 4)

C. (4, -3)

D. (3, 5)

Answer: **Option A**

Solution: Review: Solution for Number 32

**Problem 33: EE Board October 1997**

Find the major axis of the ellipse x^{2} + 4y^{2} โ 2x โ 8y + 1 = 0.

A. 2

B. 10

C. 4

D. 6

Answer: **Option C**

Solution: Review: Solution for Number 33

**Problem 34: CE Board May 1993**

The length of the latus rectum for the ellipse is equal to?

A. 2

B. 3

C. 4

D. 5

Answer: **Option C**

Solution: Review: Solution for Number 34

**Problem 35:**

An ellipse with an eccentricity of 0.65 and has one of its foci 2 units from the center. The length of the latus rectum is nearest to?

A. 3.5 units

B. 3.8 units

C. 4.2 units

D. 3.2 units

Answer: **Option A**

Solution: Review: Solution for Number 35

**Problem 36:**

An earth satellite has an apogee of 40,000 km and a perigee of 6,600 km. Assuming the radius of the earth as 6,400 km, what will be the eccentricity of the elliptical path described by the satellite with the center of the earth at one of the foci?

A. 0.46

B. 0.49

C. 0.52

D. 0.56

Answer: **Option D**

Solution: Review: Solution for Number 36

**Problem 37: ECE Board April 1998**

The major axis of the elliptical path in which the earth moves around the sun is approximately 168,000,000 miles and the eccentricity of the ellipse is 1/60. Determine the apogee of the earth.

A. 93,000,000 miles

B. 91,450,000 miles

C. 94,335,100 miles

D. 94,550,000 miles

Answer: **Option D**

Solution: Review: Solution for Number 37

**Problem 38: CE Board November 1992**

The earthโs orbit is an ellipse with the sun at one of the foci. If the farthest distance of the sun from the earth is 105.5 million km and the nearest distance of the sun from the earth is 78.25 million km, find the eccentricity of the ellipse.

A. 0.15

B. 0.25

C. 0.35

D. 0.45

Answer: **Option A**

Solution: Review: Solution for Number 38

**Problem 39:**

An ellipse with center at the origin has a length of major axis 20 units. If the distance from center of ellipse to its focus is 5, what is the equation of its directrix?

A. x = 18

B. x = 20

C. x = 15

D. x = 16

Answer: **Option B**

Solution: Review: Solution for Number 39

**Problem 40:**

What is the length of the latus rectum of 4x^{2} + 9y^{2} + 8x โ 32 = 0?

A. 2.5

B. 2.7

C. 2.3

D. 2.9

Answer: **Option B**

Solution: Review: Solution for Number 40

**Problem 41: EE Board October 1993**

4x^{2} โ y^{2} = 16 is the equation of a/an?

A. parabola

B. hyperbola

C. circle

D. ellipse

Answer: **Option B**

Solution: Review: Solution for Number 41

**Problem 42: EE Board October 1993**

Find the eccentricity of the curve 9x^{2} โ 4y^{2} โ 36x + 8y = 4.

A. 1.80

B. 1.92

C. 1.86

D. 1.76

Answer: **Option A**

Solution: Review: Solution for Number 42

**Problem 43: CE Board November 1995**

How far from the x-axis is the focus F of the hyperbola x^{2} โ 2y^{2} + 4x + 4y + 4 = 0?

A. 4.5

B. 3.4

C. 2.7

D. 2.1

Answer: **Option C**

Solution: Review: Solution for Number 43

**Problem 44: EE Board October 1994**

The semi-transverse axis of the hyperbola is?

A. 2

B. 3

C. 4

D. 5

Answer: **Option B**

Solution: Review: Solution for Number 44

**Problem 45: CE Board May 1996**

What is the equation of the asymptote of the hyperbola?

A. 2x โ 3y = 0

B. 3x โ 2y = 0

C. 2x โ y = 0

D. 2x + y = 0

Answer: **Option A**

Solution: Review: Solution for Number 45

**Problem 46: EE Board April 1994**

Find the equation of the hyperbola whose asymptotes are y = ยฑ 2x and which passes through (5/2, 3).

A. 4x^{2} + y^{2} + 16 = 0

B. 4x^{2} + y^{2} โ 16 = 0

C. x^{2} โ 4y^{2} โ 16 = 0

D. 4x^{2} โ y^{2} = 16

Answer: **Option D**

Solution: Review: Solution for Number 46

**Problem 47:**

Find the equation of the hyperbola with vertices (-4, 2) and (0, 2) and foci (-5, 2) and (1, 2).

A. 5x^{2} โ 4y^{2} + 20x +16y โ 16 = 0

B. 5x^{2} โ 4y^{2} + 20x โ 16y โ 16 = 0

C. 5x^{2} โ 4y^{2} โ 20x +16y + 16 = 0

D. 5x^{2} + 4y^{2} โ 20x +16y โ 16 = 0

Answer: **Option A**

Solution: Review: Solution for Number 47

**Problem 48:**

Find the distance between P1 (6, -2, -3) and P2 (5, 1, -4).

A. 11

B. โ11

C. 12

D. โ12

Answer: **Option B**

Solution: Review: Solution for Number 48

**Problem 49:**

The point of intersection of the planes x + 5y โ 2z = 9; 3x โ 2y + z = 3 and x + y + z = 2 is at?

A. (2, 1, -1)

B. (2, 0, -1)

C. (-1, 1, -1)

D. (-1, 2, -1)

Answer: **Option A**

Solution: Review: Solution for Number 49

**Problem 50: ME Board April 1997**

What is the radius of the sphere center at the origin that passes the point 8, 1, 6?

A. 10

B. 9

C. โ101

D. 10.5

Answer: **Option C**

Solution: Review: Solution for Number 50

**Problem 51:**

The equation of a sphere with center at (-3, 2, 4) and of radius 6 units is?

A. x^{2} + y^{2} + z^{2} +6x โ 4y โ 8z = 36

B. x^{2} + y^{2} + z^{2} +6x โ 4y โ 8z = 7

C. x^{2} + y^{2} + z^{2} +6x โ 4y + 8z = 6

D. x^{2} + y^{2} + z^{2} +6x โ 4y + 8z = 36

Answer: **Option B**

Solution: Review: Solution for Number 51

**Problem 52: EE Board April 1997**

Find the polar question of the circle, if its center is at (4, 0) and the radius 4.

A. r โ 8 cos ฮธ = 0

B. r โ 6 cos ฮธ = 0

C. r โ 12 cos ฮธ = 0

D. r โ 4 cos ฮธ = 0

Answer: **Option A**

Solution: Review: Solution for Number 52

**Problem 53: ME Board October 1996**

What are the x and y coordinates of the focus of the iconic section described by the following equation? (Angle ฮธ corresponds to a right triangle with adjacent side x, opposite side y and the hypotenuse r.)ย **r sin ^{2} ฮธ = cos ฮธ**

A. (1/4, 0)

B. (0, ฯ/2)

C. (0, 0)

D. (-1/2, 0)

Answer: **Option A**

Solution: Review: Solution for Number 53

**Problem 54:**

Find the polar equation of the circle of radius 3 units and center at (3, 0).

A. r = 3 cos ฮธ

B. r = 3 sin ฮธ

C. r = 6 cos ฮธ

D. r = 9 sin ฮธ

Answer: **Option C**

Solution: Review: Solution for Number 54

**Problem 55: EE Board October 1997**

Given the polar equation r = 5 sin ฮธ. Determine the rectangular coordinate (x, y) of a point in the curve when ฮธ is 30ยบ.

A. (2.17, 1.25)

B. (3.08, 1.5)

C. (2.51, 4.12)

D. (6, 3)

Answer: **Option A**

Solution: Review: Solution for Number 55

### Online Questions and Answers in Analytic Geometry: Parabola, Ellipse and Hyperbola Series

Following is the list of multiple choice questions in this brand new series:

**MCQ in Analytic Geometry: Parabola, Ellipse and Hyperbola**

**MCQ from Number 1 โ 50**Answer key:

**PART 1**

**MCQ from Number 51 โ 100**Answer key:

**PART 2**

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The given in Problem 7 looks incorrect especially the 2nd equation of the circle. Please give this a notice