Home › Board exam reviewer › Analytic geometry

Analytic geometry: board exam questions with solutions

On the ECE exam under Mathematics and the REE exam under Mathematics. Try each question before you open its solution, then practise with fresh numbers below: 2 computed question types on this topic.

Loading the reviewer… (it needs JavaScript)

Questions

1. What is the distance between the points (1, 2) and (4, 6)?

  1. 5
  2. 7
  3. 25
  4. √7
Show the answer and solution

Answer: A. 5

d = √[(4 − 1)² + (6 − 2)²] = √(9 + 16) = √25 = 5.

2. What is the slope of the line through (−2, 3) and (4, −9)?

  1. −2
  2. 2
  3. −1/2
  4. −6
Show the answer and solution

Answer: A. −2

m = (−9 − 3)/(4 − (−2)) = −12/6 = −2.

3. Find the perpendicular distance from the point (3, 4) to the line 3x + 4y − 10 = 0.

  1. 5
  2. 3
  3. 15
  4. 2
Show the answer and solution

Answer: B. 3

d = |3(3) + 4(4) − 10|/√(3² + 4²) = |15|/5 = 3.

4. Find the center and radius of the circle x² + y² − 6x + 8y − 11 = 0.

  1. Center (−3, 4), r = 6
  2. Center (3, −4), r = 4
  3. Center (−3, 4), r = √11
  4. Center (3, −4), r = 6
Show the answer and solution

Answer: D. Center (3, −4), r = 6

Complete the squares: (x − 3)² + (y + 4)² = 11 + 9 + 16 = 36. So the center is (3, −4) and r = 6.

5. What is the focus of the parabola y² = 12x?

  1. (12, 0)
  2. (0, 3)
  3. (3, 0)
  4. (6, 0)
Show the answer and solution

Answer: C. (3, 0)

Compare with y² = 4ax: 4a = 12, so a = 3 and the focus is at (a, 0) = (3, 0).

6. What is the eccentricity of the ellipse x²/25 + y²/9 = 1?

  1. 0.6
  2. 0.8
  3. 1.25
  4. 0.36
Show the answer and solution

Answer: B. 0.8

a = 5 and b = 3, so c = √(25 − 9) = 4 and e = c/a = 4/5 = 0.8.

7. For any hyperbola, the eccentricity e satisfies which condition?

  1. e = 1
  2. 0 < e < 1
  3. e = 0
  4. e > 1
Show the answer and solution

Answer: D. e > 1

Hyperbolas have e = c/a with c > a, so e > 1. A parabola has e = 1, an ellipse 0 < e < 1, and a circle e = 0.

8. Find the line through (1, 2) that is perpendicular to 2x − 3y + 5 = 0.

  1. 2x − 3y + 4 = 0
  2. 3x − 2y + 1 = 0
  3. 3x + 2y − 7 = 0
  4. 3x + 2y + 7 = 0
Show the answer and solution

Answer: C. 3x + 2y − 7 = 0

The given line has slope 2/3, so the perpendicular slope is −3/2. y − 2 = −(3/2)(x − 1) rearranges to 3x + 2y − 7 = 0. 2x − 3y + 4 = 0 passes through the point but is parallel.

Worked examples with computed numbers

These numbers are fixed on this page; in practice they change every time.

9. What is the radius of the circle x² + y² − 8x + 14y + 1 = 0?

  1. 8.062
  2. 64
  3. 8
  4. 1
Show the answer and solution

Answer: C. 8

Complete the squares: centre (4, -7), r² = h² + k² − F = 65 − (1) = 64, so r = 8.

10. Find the distance from the point (-6, -6) to the line 5x − 7y + 19 = 0.

  1. 31
  2. 3.604
  3. 2.583
  4. 1.395
Show the answer and solution

Answer: B. 3.604

d = |Ax₀ + By₀ + C|/√(A² + B²) = |31|/√74 = 3.604.

Other topics in the same subjects

Algebra · Trigonometry · Differential calculus · Integral calculus · Differential equations · Complex numbers · Probability and statistics · Matrices and determinants · Laplace transforms and Fourier series

An independent, free reviewer: not affiliated with the PRC or the Boards. Subject weights and passing rules are from the PRC examination programs and the laws (RA 9292, RA 7920); check the current PRC program before you file. Code questions follow the Philippine Electrical Code 2017 Part 1; check them against your own copy of the Code.