NBT NBT Mathematics: Analytical Geometry — Questions and Answers
Question 1: What is the distance between the points (-2, 3) and (4, -5)?
- 8
- 10 (Correct answer)
- 14
- √52
Correct answer: 10
d = √((4+2)² + (-5-3)²) = √(36+64) = √100 = 10.
Question 2: What is the midpoint of the segment joining (3, -7) and (-5, 1)?
- (4, -4)
- (-4, 4)
- (-1, -3) (Correct answer)
- (1, 3)
Correct answer: (-1, -3)
Midpoint = ((3−5)/2, (−7+1)/2) = (−1, −3).
Question 3: M(2, 5) is the midpoint of AB, where A is (-1, 8). What are the coordinates of B?
- (0.5, 6.5)
- (3, -3)
- (-4, 11)
- (5, 2) (Correct answer)
Correct answer: (5, 2)
B = (2·2 − (−1), 2·5 − 8) = (5, 2).
Question 4: What is the slope of a line perpendicular to 3x − 4y = 12?
- -4/3 (Correct answer)
- 3/4
- -3/4
- 4/3
Correct answer: -4/3
The given line has slope 3/4, so a perpendicular line has slope −4/3.
Question 5: Which equation describes the line through (1, 2) parallel to y = −2x + 7?
- y = 2x
- y = −2x + 4 (Correct answer)
- y = −2x + 2
- y = ½x + 3/2
Correct answer: y = −2x + 4
Parallel lines share slope −2; substituting (1, 2) gives c = 4.
Question 6: For what value of k are the points (1, 1), (3, 5) and (k, 9) collinear?
Correct answer: 5
The slope from (1,1) to (3,5) is 2, so 4/(k−3) = 2 gives k = 5.
Question 7: What is the y-intercept of the line 2x + 5y = 20?
- 10
- -4
- 5/2
- 4 (Correct answer)
Correct answer: 4
Setting x = 0 gives 5y = 20, so y = 4.
Question 8: What are the center and radius of the circle x² + y² − 6x + 4y − 12 = 0?
- Center (3, -2), radius 5 (Correct answer)
- Center (-3, 2), radius 5
- Center (3, -2), radius √12
- Center (-6, 4), radius 25
Correct answer: Center (3, -2), radius 5
Completing the square gives (x−3)² + (y+2)² = 25.
Question 9: Which is the equation of the circle with center (-1, 4) and radius 3?
- (x − 1)² + (y + 4)² = 9
- (x + 1)² + (y − 4)² = 9 (Correct answer)
- (x + 1)² + (y − 4)² = 3
- (x − 1)² + (y − 4)² = 9
Correct answer: (x + 1)² + (y − 4)² = 9
Standard form (x−a)² + (y−b)² = r² with a = −1, b = 4, r² = 9.
Question 10: A circle has a diameter with endpoints (2, 1) and (6, 7). What is its equation?
- (x − 4)² + (y − 4)² = 52
- (x − 4)² + (y − 4)² = √13
- (x − 4)² + (y − 4)² = 13 (Correct answer)
- (x − 2)² + (y − 3)² = 13
Correct answer: (x − 4)² + (y − 4)² = 13
Center is the midpoint (4, 4) and r² = (6−4)² + (7−4)² = 13.
Question 11: Where does the point (5, 1) lie relative to the circle x² + y² = 25?
- Inside the circle
- On the circle
- At the center
- Outside the circle (Correct answer)
Correct answer: Outside the circle
5² + 1² = 26 > 25, so the point is outside.
Question 12: What is the equation of the tangent to x² + y² = 25 at the point (3, 4)?
- 3x + 4y = 25 (Correct answer)
- 4x + 3y = 25
- 3x − 4y = 25
- 4x − 3y = 7
Correct answer: 3x + 4y = 25
The radius has slope 4/3, so the tangent has slope −3/4 through (3, 4), giving 3x + 4y = 25.
Question 13: A circle centered at (2, -3) touches the x-axis. What is its radius?
Correct answer: 3
The radius equals the distance from the center to the x-axis, |−3| = 3.
Question 14: For what value of k does x² + y² + 4x − 2y + k = 0 represent a circle of radius 3?
- 4
- -4 (Correct answer)
- 14
- -14
Correct answer: -4
Completing the square gives (x+2)² + (y−1)² = 5 − k, so 5 − k = 9 and k = −4.
Question 15: What is the angle of inclination of the line y = √3x + 2?
- 30°
- 45°
- 120°
- 60° (Correct answer)
Correct answer: 60°
tan θ = √3, so θ = 60°.
Question 16: What is the angle of inclination of the line x + y = 5?
- 135° (Correct answer)
- 45°
- -45°
- 225°
Correct answer: 135°
The slope is −1, and tan θ = −1 for 0° ≤ θ < 180° gives θ = 135°.
Question 17: What is the perpendicular distance from (1, 2) to the line 3x + 4y − 1 = 0?
Correct answer: 2
d = |3(1) + 4(2) − 1| / √(3² + 4²) = 10/5 = 2.
Question 18: What is the area of the triangle with vertices (0, 0), (6, 0) and (0, 8)?
- 48
- 24 (Correct answer)
- 14
- 10
Correct answer: 24
It is a right triangle with legs 6 and 8, so area = ½ × 6 × 8 = 24.
Question 19: ABCD is a parallelogram with A(1, 2), B(5, 3) and C(6, 7). What are the coordinates of D?
- (10, 8)
- (0, -2)
- (3, 5)
- (2, 6) (Correct answer)
Correct answer: (2, 6)
Diagonals bisect each other, so D = A + C − B = (2, 6).
Question 20: Triangle PQR has vertices P(1, 1), Q(4, 5) and R(1, 5). At which vertex is the right angle?
- P
- Q
- R (Correct answer)
- There is no right angle
Correct answer: R
PR is vertical and RQ is horizontal, so they meet at a right angle at R.
Question 21: What is the acute angle between the lines y = x and y = √3x?
- 15° (Correct answer)
- 30°
- 45°
- 105°
Correct answer: 15°
Their inclinations are 45° and 60°, so the angle between them is 15°.
