[ DOST 12 ] DOST SCHOLARSHIP EXAM REVIEWER XII

Analytic Geometry

Instructions:

Please answer each question to the best of your ability. Each question is multiple choice, and only one answer per question is correct. Select the most appropriate answer from the options provided. There are 20 questions in total.

When you have completed all questions, click the "Submit" button at the bottom of the page to see your score. Good luck!

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

A) 3
B) 5
C) 7
D) 6

2. Find the midpoint of the line segment joining the points \( (2, 3) \) and \( (8, 7) \).

A) \( (4, 5) \)
B) \( (5, 5) \)
C) \( (3, 4) \)
D) \( (7, 5) \)

3. Which of the following is the equation of a circle with center \( (3, -2) \) and radius 5?

A) \( (x - 3)^2 + (y + 2)^2 = 25 \)
B) \( (x + 3)^2 + (y - 2)^2 = 25 \)
C) \( (x - 3)^2 + (y - 2)^2 = 5 \)
D) \( (x + 3)^2 + (y + 2)^2 = 5 \)

4. What is the slope of the line passing through the points \( (1, 2) \) and \( (3, 6) \)?

A) 2
B) 3
C) 4
D) 1

5. Determine the equation of the line with slope \( \frac{3}{4} \) passing through the point \( (2, -1) \).

A) \( y = \frac{3}{4}x - \frac{5}{2} \)
B) \( y = \frac{3}{4}x + 2 \)
C) \( y = -\frac{3}{4}x - 2 \)
D) \( y = \frac{4}{3}x - 1 \)

6. Find the length of the line segment joining \( (3, 4) \) and \( (6, 8) \).

A) 4
B) 5
C) 7
D) 6

7. What is the general form of the equation of a line passing through \( (4, -3) \) with slope -2?

A) \( y = -2x + 3 \)
B) \( y = -2x + 5 \)
C) \( y = 2x - 3 \)
D) \( y = x - 3 \)

8. Determine the equation of the perpendicular bisector of the line segment joining \( (1, 1) \) and \( (5, 5) \).

A) \( y = x + 3 \)
B) \( y = -x + 6 \)
C) \( y = 2x - 1 \)
D) \( y = x - 1 \)

9. What is the radius of a circle with equation \( x^2 + y^2 - 4x + 6y = 12 \)?

A) 5
B) 6
C) 4
D) 3

10. Find the slope of a line perpendicular to \( y = \frac{1}{3}x + 4 \).

A) \( \frac{1}{3} \)
B) -3
C) 3
D) \( -\frac{1}{3} \)

11. Which of the following points is on the line \( y = 2x - 5 \)?

A) (3, 1)
B) (2, 3)
C) (1, 1)
D) (5, 5)

12. What is the equation of a parabola with vertex \( (0, 0) \) that opens upwards and has a focus at \( (0, 2) \)?

A) \( y = \frac{1}{4}x^2 \)
B) \( y = 4x^2 \)
C) \( x = 4y^2 \)
D) \( y = 2x^2 \)

13. Find the eccentricity of an ellipse with equation \( \frac{x^2}{16} + \frac{y^2}{9} = 1 \).

A) \( \frac{1}{2} \)
B) 1
C) \( \frac{3}{4} \)
D) \( \frac{5}{4} \)

14. What is the distance between the parallel lines \( y = 2x + 1 \) and \( y = 2x - 3 \)?

A) 2
B) 3
C) 4
D) 5

15. Find the equation of the ellipse with foci at \( (0, \pm5) \) and vertices at \( (0, \pm7) \).

A) \( \frac{x^2}{24} + \frac{y^2}{49} = 1 \)
B) \( \frac{x^2}{36} + \frac{y^2}{49} = 1 \)
C) \( \frac{x^2}{25} + \frac{y^2}{49} = 1 \)
D) \( \frac{x^2}{16} + \frac{y^2}{25} = 1 \)

16. A line has the equation \( 3x + 4y = 12 \). What is the x-intercept of this line?

A) -3
B) 4
C) 6
D) 3

17. Find the coordinates of the focus of the parabola with equation \( y^2 = 8x \).

A) (0, 4)
B) (0, -4)
C) (2, 0)
D) (-2, 0)

18. If a line has a slope of \( \frac{2}{3} \) and passes through the point \( (3, -1) \), which of the following is its equation?

A) \( y = \frac{2}{3}x - 3 \)
B) \( y = \frac{3}{2}x + 3 \)
C) \( y = -\frac{2}{3}x + 1 \)
D) \( y = \frac{3}{2}x - 2 \)

19. Find the equation of the hyperbola with vertices at \( (\pm4, 0) \) and foci at \( (\pm5, 0) \).

A) \( \frac{x^2}{25} - \frac{y^2}{9} = 1 \)
B) \( \frac{x^2}{16} - \frac{y^2}{9} = 1 \)
C) \( \frac{x^2}{9} - \frac{y^2}{16} = 1 \)
D) \( \frac{x^2}{25} - \frac{y^2}{16} = 1 \)

20. What is the equation of the line passing through \( (-2, 1) \) and perpendicular to the line \( y = -\frac{1}{2}x + 4 \)?

A) \( y = -2x - 3 \)
B) \( y = \frac{1}{2}x - 1 \)
C) \( y = 2x + 5 \)
D) \( y = 2x - 1 \)

Result

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