NBT Mathematics: Analytical Geometry Flashcards

What is the distance between the points (-2, 3) and (4, -5)?

Answer: 10

d = √((4+2)² + (-5-3)²) = √(36+64) = √100 = 10.

What is the midpoint of the segment joining (3, -7) and (-5, 1)?

Answer: (-1, -3)

Midpoint = ((3−5)/2, (−7+1)/2) = (−1, −3).

M(2, 5) is the midpoint of AB, where A is (-1, 8). What are the coordinates of B?

Answer: (5, 2)

B = (2·2 − (−1), 2·5 − 8) = (5, 2).

What is the slope of a line perpendicular to 3x − 4y = 12?

Answer: -4/3

The given line has slope 3/4, so a perpendicular line has slope −4/3.

Which equation describes the line through (1, 2) parallel to y = −2x + 7?

Answer: y = −2x + 4

Parallel lines share slope −2; substituting (1, 2) gives c = 4.

For what value of k are the points (1, 1), (3, 5) and (k, 9) collinear?

Answer: 5

The slope from (1,1) to (3,5) is 2, so 4/(k−3) = 2 gives k = 5.

What is the y-intercept of the line 2x + 5y = 20?

Answer: 4

Setting x = 0 gives 5y = 20, so y = 4.

What are the center and radius of the circle x² + y² − 6x + 4y − 12 = 0?

Answer: Center (3, -2), radius 5

Completing the square gives (x−3)² + (y+2)² = 25.

Which is the equation of the circle with center (-1, 4) and radius 3?

Answer: (x + 1)² + (y − 4)² = 9

Standard form (x−a)² + (y−b)² = r² with a = −1, b = 4, r² = 9.

A circle has a diameter with endpoints (2, 1) and (6, 7). What is its equation?

Answer: (x − 4)² + (y − 4)² = 13

Center is the midpoint (4, 4) and r² = (6−4)² + (7−4)² = 13.

Where does the point (5, 1) lie relative to the circle x² + y² = 25?

Answer: Outside the circle

5² + 1² = 26 > 25, so the point is outside.

What is the equation of the tangent to x² + y² = 25 at the point (3, 4)?

Answer: 3x + 4y = 25

The radius has slope 4/3, so the tangent has slope −3/4 through (3, 4), giving 3x + 4y = 25.

A circle centered at (2, -3) touches the x-axis. What is its radius?

Answer: 3

The radius equals the distance from the center to the x-axis, |−3| = 3.

For what value of k does x² + y² + 4x − 2y + k = 0 represent a circle of radius 3?

Answer: -4

Completing the square gives (x+2)² + (y−1)² = 5 − k, so 5 − k = 9 and k = −4.

What is the angle of inclination of the line y = √3x + 2?

Answer: 60°

tan θ = √3, so θ = 60°.

What is the angle of inclination of the line x + y = 5?

Answer: 135°

The slope is −1, and tan θ = −1 for 0° ≤ θ < 180° gives θ = 135°.

What is the perpendicular distance from (1, 2) to the line 3x + 4y − 1 = 0?

Answer: 2

d = |3(1) + 4(2) − 1| / √(3² + 4²) = 10/5 = 2.

What is the area of the triangle with vertices (0, 0), (6, 0) and (0, 8)?

Answer: 24

It is a right triangle with legs 6 and 8, so area = ½ × 6 × 8 = 24.

ABCD is a parallelogram with A(1, 2), B(5, 3) and C(6, 7). What are the coordinates of D?

Answer: (2, 6)

Diagonals bisect each other, so D = A + C − B = (2, 6).

Triangle PQR has vertices P(1, 1), Q(4, 5) and R(1, 5). At which vertex is the right angle?

Answer: R

PR is vertical and RQ is horizontal, so they meet at a right angle at R.