Coordinate Geometry for Competitive Exam Success

Coordinate Geometry

Distance Formula Ellipse Basics Equation of Circle Hyperbola Basics Parabola Basics Section Formula Straight Line

Q. If a line has the equation y = -3x + 6, what is the y-intercept?

A. 6
B. 3
C. 0
D. -3

Solution

The y-intercept is the constant term in the equation, which is 6.

Correct Answer: A — 6

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Q. If a parabola has its vertex at (3, -2) and opens downwards, what is the general form of its equation?

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

Solution

For a downward-opening parabola with vertex (h, k), the equation is y - k = -a(x - h)^2. Here, h = 3 and k = -2.

Correct Answer: B — y + 2 = -a(x - 3)^2

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Q. If a parabola opens to the left, which of the following is its standard form?

A. y^2 = -4px
B. x^2 = -4py
C. y^2 = 4px
D. x^2 = 4py

Solution

The standard form of a parabola that opens to the left is y^2 = -4px.

Correct Answer: A — y^2 = -4px

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Q. If a parabola opens to the right and has its vertex at the origin, what is the general form of its equation?

A. y^2 = 4px
B. x^2 = 4py
C. y = mx + c
D. x = ay^2

Solution

The general form of a parabola that opens to the right is y^2 = 4px.

Correct Answer: A — y^2 = 4px

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Q. If the coordinates of a point are (x, y) and it lies on the line 2x + 3y = 12, what is the value of y when x = 2?

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

Solution

Substituting x = 2: 2(2) + 3y = 12 => 4 + 3y = 12 => 3y = 8 => y = 8/3 ≈ 2.67, closest is 3.

Correct Answer: B — 3

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Q. If the coordinates of a point are (x, y) and it lies on the line 2x + 3y = 6, what is the value of y when x = 0? (2023)

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

Solution

Substituting x = 0 in 2(0) + 3y = 6 gives 3y = 6, so y = 2.

Correct Answer: B — 3

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Q. If the coordinates of a point are (x, y) and it lies on the line 3x + 4y = 12, what is y when x = 0?

A. 3
B. 4
C. 2
D. 0

Solution

Substituting x = 0: 3(0) + 4y = 12 => 4y = 12 => y = 3.

Correct Answer: B — 4

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Q. If the coordinates of point A are (1, 2) and point B are (4, 6), what is the slope of line AB?

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

Solution

Slope = (6-2)/(4-1) = 4/3.

Correct Answer: A — 2

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Q. If the coordinates of point A are (2, 3) and point B are (4, 7), what is the slope of line AB?

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

Solution

Slope = (7-3)/(4-2) = 4/2 = 2.

Correct Answer: A — 2

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Q. If the coordinates of the vertices of a triangle are (1, 2), (4, 6), and (7, 2), what is the length of the side opposite to the vertex (4, 6)? (2023)

A. 5
B. 6
C. 7
D. 8

Solution

Length = √[(1-7)² + (2-2)²] = √[36] = 6.

Correct Answer: B — 6

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Q. If the focus of a parabola is at (0, 2) and the directrix is y = -2, what is the equation of the parabola?

A. x^2 = 8y
B. x^2 = 4y
C. y^2 = 8x
D. y^2 = 4x

Solution

The distance from the focus to the directrix is 4, so the equation is x^2 = 8y.

Correct Answer: A — x^2 = 8y

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Q. If the line 2x - 3y + 6 = 0 intersects the x-axis, what is the x-coordinate of the intersection point?

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

Solution

Set y = 0: 2x + 6 = 0; x = -3.

Correct Answer: A — -3

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Q. If the line 4x + 3y = 12 intersects the y-axis, what is the point of intersection? (2022)

A. (0, 4)
B. (0, 3)
C. (0, 2)
D. (0, 1)

Solution

Setting x = 0 in the equation gives 3y = 12, thus y = 4. The point of intersection is (0, 4).

Correct Answer: B — (0, 3)

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Q. If the line 4x - 3y + 12 = 0 is parallel to another line, what is the slope of the parallel line? (2022)

A. 4/3
B. 3/4
C. -4/3
D. -3/4

Solution

Rearranging gives y = (4/3)x + 4. The slope is -4/3, so a parallel line has the same slope.

Correct Answer: C — -4/3

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Q. If the line y = mx + c passes through the point (1, 2), what is the value of c when m = 3? (2023)

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

Solution

Substituting (1, 2) gives 2 = 3(1) + c, so c = -1.

Correct Answer: A — -1

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Q. If the line y = mx + c passes through the point (2, 3) and has a slope of 2, what is the value of c? (2023)

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

Solution

Using y = mx + c: 3 = 2(2) + c => c = -1.

Correct Answer: A — -1

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Q. If the point (x, y) lies on the line 3x - 4y = 12, what is the value of y when x = 4? (2023)

A. 0
B. 3
C. 4
D. 2

Solution

Substituting x = 4 gives 3(4) - 4y = 12 => 12 - 4y = 12 => y = 0.

Correct Answer: D — 2

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Q. If the point (x, y) lies on the line 4x - 5y = 20, what is the value of y when x = 0?

A. 4
B. 5
C. 0
D. -4

Solution

Substituting x = 0: 4(0) - 5y = 20 => -5y = 20 => y = -4.

Correct Answer: B — 5

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Q. The coordinates of the centroid of a triangle with vertices at (1, 2), (3, 4), and (5, 6) are:

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

Solution

Centroid = ((1+3+5)/3, (2+4+6)/3) = (3, 4).

Correct Answer: A — (3, 4)

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Q. The coordinates of the centroid of a triangle with vertices at (2, 3), (4, 5), and (6, 7) are:

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

Solution

Centroid = ((2+4+6)/3, (3+5+7)/3) = (4, 5).

Correct Answer: B — (3, 4)

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Q. The coordinates of the foot of the perpendicular from the point (1, 2) to the line 2x + 3y = 6 are:

A. (2, 0)
B. (0, 2)
C. (1, 1)
D. (0, 0)

Solution

Using the formula for foot of perpendicular, we find the coordinates to be (2, 0).

Correct Answer: A — (2, 0)

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Q. The coordinates of the foot of the perpendicular from the point (1, 2) to the line 3x + 4y = 12 are:

A. (2, 1)
B. (1, 2)
C. (0, 3)
D. (3, 0)

Solution

Using the formula for foot of perpendicular, we find the coordinates to be (2, 1).

Correct Answer: A — (2, 1)

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Q. The coordinates of the foot of the perpendicular from the point (3, 4) to the line 2x + 3y = 6 are:

A. (2, 0)
B. (1, 2)
C. (0, 2)
D. (2, 2)

Solution

Using the formula for foot of perpendicular, we find the coordinates to be (2, 0).

Correct Answer: A — (2, 0)

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Q. The coordinates of the foot of the perpendicular from the point (3, 4) to the line 2x + 3y - 6 = 0 are:

A. (2, 0)
B. (0, 2)
C. (1, 1)
D. (2, 2)

Solution

Using the formula for foot of perpendicular, we find the coordinates to be (2, 0).

Correct Answer: A — (2, 0)

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Q. The equation of a line with slope 2 passing through the point (1, 3) is?

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

Solution

Using point-slope form: y - 3 = 2(x - 1) => y = 2x + 1.

Correct Answer: C — y = 2x + 3

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Q. The equation of a parabola with vertex at (0, 0) and directrix y = -3 is?

A. x^2 = -12y
B. y^2 = -12x
C. x^2 = 12y
D. y^2 = 12x

Solution

The distance from the vertex to the directrix is 3, so the equation is x^2 = -12y.

Correct Answer: A — x^2 = -12y

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Q. The equation of a parabola with vertex at (0, 0) and focus at (0, 3) is?

A. x^2 = 12y
B. y^2 = 12x
C. x^2 = 6y
D. y^2 = 6x

Solution

The distance from the vertex to the focus is 3, so the equation is x^2 = 12y.

Correct Answer: A — x^2 = 12y

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Q. The parabola y^2 = 12x opens in which direction?

A. Upwards
B. Downwards
C. Left
D. Right

Solution

The equation y^2 = 12x indicates that the parabola opens to the right since it is in the form y^2 = 4px.

Correct Answer: D — Right

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Q. The vertex of the parabola given by the equation y = 2x^2 - 4x + 1 is located at which point?

A. (1, -1)
B. (2, 0)
C. (1, 0)
D. (0, 1)

Solution

To find the vertex, use the formula x = -b/(2a). Here, a = 2, b = -4, so x = 1. Plugging x = 1 into the equation gives y = -1.

Correct Answer: A — (1, -1)

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Q. What is the angle between the lines 2x + 3y = 6 and 4x - y = 5?

A. 45 degrees
B. 60 degrees
C. 90 degrees
D. 30 degrees

Solution

The slopes of the lines are -2/3 and 4. The angle θ can be found using tan(θ) = |(m1 - m2) / (1 + m1*m2)|.

Correct Answer: B — 60 degrees

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Coordinate Geometry

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