Coordinate Geometry - Distance & Section Formula for Exams

Coordinate Geometry - Distance and Section Formula - Coordinate Geometry Applications - Case Studies

Q. If a circle is centered at (2, 3) with a radius of 5, what is the equation of the circle?

A. (x - 2)² + (y - 3)² = 25
B. (x + 2)² + (y + 3)² = 25
C. (x - 2)² + (y + 3)² = 25
D. (x + 2)² + (y - 3)² = 25

Solution

The standard form of a circle's equation is (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius.

Correct Answer: A — (x - 2)² + (y - 3)² = 25

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Q. If a line has the equation y = 3x + 2, what is the y-intercept?

A. 2
B. 3
C. 1
D. 0

Solution

The y-intercept is the value of y when x = 0, which is 2.

Correct Answer: A — 2

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Q. If point A(2, 3) and point B(8, 7) are endpoints of a line segment, what is the midpoint M?

A. (5, 5)
B. (4, 5)
C. (6, 5)
D. (5, 4)

Solution

Midpoint M = ((x1 + x2)/2, (y1 + y2)/2) = ((2 + 8)/2, (3 + 7)/2) = (5, 5).

Correct Answer: A — (5, 5)

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Q. If point C is at (5, 5) and point D is at (5, 10), what is the distance between points C and D?

A. 5
B. 10
C. 15
D. 20

Solution

Distance = |y2 - y1| = |10 - 5| = 5.

Correct Answer: A — 5

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Q. If point C(1, 1) divides the line segment joining points A(0, 0) and B(4, 4) in the ratio k:1, what is the value of k?

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

Solution

Using the section formula, C = ((kx2 + x1)/(k+1), (ky2 + y1)/(k+1)). Setting C(1, 1) gives k = 1.

Correct Answer: B — 2

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Q. If the coordinates of point D are (5, 5) and it is the midpoint of line segment joining points E(3, 1) and F(x, y), what are the coordinates of F?

A. (7, 9)
B. (9, 7)
C. (8, 6)
D. (6, 8)

Solution

Using the midpoint formula: D = ((x1 + x2)/2, (y1 + y2)/2). Solving gives F(7, 9).

Correct Answer: B — (9, 7)

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Q. What is the coordinates of the point that divides the line segment joining (2, 3) and (4, 7) in the ratio 1:3?

A. (3, 5)
B. (2.5, 4)
C. (3.5, 5.5)
D. (3, 6)

Solution

Using the section formula: P = ((1*4 + 3*2)/(1+3), (1*7 + 3*3)/(1+3)) = (3, 5).

Correct Answer: A — (3, 5)

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Q. What is the distance between the points (-2, -3) and (2, 1)?

A. 5.66
B. 6.32
C. 4.47
D. 5.0

Solution

Using the distance formula: d = √((2 - (-2))² + (1 - (-3))²) = √(16 + 16) = √32 = 4√2 ≈ 5.66.

Correct Answer: A — 5.66

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Q. What is the distance between the points (0, 0) and (x, y) if the distance is 10?

A. x² + y² = 100
B. x² + y² = 10
C. x² + y² = 50
D. x² + y² = 25

Solution

Using the distance formula: d = √(x² + y²) = 10 implies x² + y² = 100.

Correct Answer: A — x² + y² = 100

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Q. What is the length of the line segment connecting the points (1, 2) and (4, 6)?

A. 5.0
B. 4.0
C. 3.0
D. 6.0

Solution

Using the distance formula: d = √((4 - 1)² + (6 - 2)²) = √(9 + 16) = √25 = 5.0.

Correct Answer: A — 5.0

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Q. What is the section formula for dividing a line segment in the ratio 2:3?

A. (2x2 + 3x1)/(2 + 3), (2y2 + 3y1)/(2 + 3)
B. (3x2 + 2x1)/(3 + 2), (3y2 + 2y1)/(3 + 2)
C. (x1 + x2)/2, (y1 + y2)/2
D. (x2 - x1)/(y2 - y1)

Solution

The section formula for dividing a line segment in the ratio m:n is ((mx2 + nx1)/(m+n), (my2 + ny1)/(m+n)).

Correct Answer: A — (2x2 + 3x1)/(2 + 3), (2y2 + 3y1)/(2 + 3)

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Q. What is the section ratio of the point (4, 5) that divides the line segment joining (2, 3) and (6, 7) internally?

A. 1:1
B. 2:1
C. 3:1
D. 1:2

Solution

Using the section formula, the ratio is 1:1 since the coordinates are equidistant from the midpoint (4, 5).

Correct Answer: B — 2:1

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Showing 1 to 12 of 12 (1 Pages)

Coordinate Geometry - Distance and Section Formula - Coordinate Geometry Applications - Case Studies

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