Coordinate Geometry - Exam Prep for Competitive Exams

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

Q. If a point P divides the line segment joining A(2, 3) and B(8, 7) in the ratio 3:1, what are the coordinates of P?

A. (5, 4)
B. (6, 5)
C. (7, 6)
D. (4, 5)

Solution

Using the section formula: P = ((3*8 + 1*2)/(3+1), (3*7 + 1*3)/(3+1)) = (6, 5).

Correct Answer: B — (6, 5)

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Q. If a point P divides the segment joining A(1, 2) and B(5, 6) in the ratio 3:1, what are the coordinates of P?

A. (3, 4)
B. (4, 5)
C. (2.5, 3.5)
D. (3.5, 4.5)

Solution

Using the section formula: P = ((mx2 + nx1)/(m+n), (my2 + ny1)/(m+n)) = ((3*5 + 1*1)/(3+1), (3*6 + 1*2)/(3+1)) = (3.5, 4.5).

Correct Answer: D — (3.5, 4.5)

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Q. If point A is at (1, 2) and point B is at (4, 6), what is the section formula for point C that divides AB in the ratio 1:2?

A. (2, 3)
B. (3, 4)
C. (2.5, 4)
D. (3.5, 5)

Solution

Using the section formula: C = ((mx2 + nx1)/(m+n), (my2 + ny1)/(m+n)) where m=1, n=2. C = ((1*4 + 2*1)/(1+2), (1*6 + 2*2)/(1+2)) = (3, 4).

Correct Answer: B — (3, 4)

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Q. If point C is at (4, 5) and point D is at (10, 15), what is the distance between C and D?

A. 10
B. 11.18
C. 12
D. 9.22

Solution

Using the distance formula: d = √((10 - 4)² + (15 - 5)²) = √(36 + 100) = √136 ≈ 11.18.

Correct Answer: B — 11.18

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Q. What is the area of a circle with a radius of 7 units?

A. 49π
B. 14π
C. 21π
D. 28π

Solution

Area = πr² = π(7)² = 49π.

Correct Answer: A — 49π

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Q. What is the area of a triangle with vertices at (0, 0), (4, 0), and (4, 3)?

A. 6
B. 12
C. 8
D. 10

Solution

Area = 0.5 * base * height = 0.5 * 4 * 3 = 6.

Correct Answer: B — 12

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Q. What is the distance from the point (2, -3) to the line 3x + 4y - 12 = 0?

A. 2.5
B. 3.0
C. 4.0
D. 5.0

Solution

Distance from point (x0, y0) to line Ax + By + C = 0 is given by |Ax0 + By0 + C| / √(A² + B²). Here, A=3, B=4, C=-12. Distance = |3*2 + 4*(-3) - 12| / √(3² + 4²) = |6 - 12 - 12| / 5 = 3.0.

Correct Answer: B — 3.0

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Q. What is the distance from the point (3, 2) to the line 2x + 3y - 6 = 0?

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

Solution

Distance from point (x0, y0) to line Ax + By + C = 0 is given by |Ax0 + By0 + C| / √(A² + B²). Here, A=2, B=3, C=-6. Distance = |2*3 + 3*2 - 6| / √(2² + 3²) = |6 + 6 - 6| / √(4 + 9) = 6 / √13 ≈ 1.67, which rounds to 2.

Correct Answer: B — 2

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Q. What is the equation of the line passing through the points (2, 3) and (4, 7)?

A. y = 2x - 1
B. y = 2x + 1
C. y = 2x + 3
D. y = 2x - 3

Solution

Slope m = (y2 - y1) / (x2 - x1) = (7 - 3) / (4 - 2) = 2. Using point-slope form: y - 3 = 2(x - 2) => y = 2x + 1.

Correct Answer: B — y = 2x + 1

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Q. What is the length of the line segment joining the points (0, 0) and (5, 12)?

A. 12.5
B. 13
C. 11
D. 10

Solution

Using the distance formula: d = √((5 - 0)² + (12 - 0)²) = √(25 + 144) = √169 = 13.

Correct Answer: B — 13

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Q. What is the radius of a circle with the equation (x - 3)² + (y + 2)² = 25?

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

Solution

The standard form of a circle is (x - h)² + (y - k)² = r². Here, r² = 25, so r = √25 = 5.

Correct Answer: A — 5

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Q. What is the ratio in which the point (3, 4) divides the line segment joining (1, 2) and (5, 6)?

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

Solution

Using the section formula, we can find the ratio by solving for m:n in the equations derived from the coordinates.

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 - Applications

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