Coordinate Geometry for Competitive Exam Success

Coordinate Geometry

Circles Conic sections: Parabola, Ellipse, Hyperbola Ellipse Family of Curves Hyperbola Pair of Straight Lines Parabola Straight lines

Q. What is the family of curves represented by the equation y = k/x?

A. Linear functions
B. Hyperbolas
C. Parabolas
D. Circles

Solution

The equation y = k/x represents a family of hyperbolas with varying asymptotes depending on the value of 'k'.

Correct Answer: B — Hyperbolas

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Q. What is the general form of the family of curves for the equation x^2 + y^2 = r^2?

A. Ellipses
B. Hyperbolas
C. Circles
D. Parabolas

Solution

The equation x^2 + y^2 = r^2 represents a family of circles with varying radii (r).

Correct Answer: C — Circles

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Q. What is the general form of the family of curves represented by the equation x^2 + y^2 = r^2?

A. Circles with radius r
B. Ellipses with semi-major axis r
C. Hyperbolas with transverse axis r
D. Straight lines with slope r

Solution

The equation x^2 + y^2 = r^2 represents a family of circles with radius r centered at the origin.

Correct Answer: A — Circles with radius r

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Q. What is the general form of the family of curves represented by the equation y = mx^2 + c?

A. Parabolas
B. Circles
C. Ellipses
D. Hyperbolas

Solution

The equation y = mx^2 + c represents a family of parabolas that open upwards or downwards depending on the sign of m.

Correct Answer: A — Parabolas

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Q. What is the general form of the family of curves represented by y^2 = 4ax?

A. Parabolas opening to the right
B. Circles with varying centers
C. Ellipses with varying foci
D. Hyperbolas with varying asymptotes

Solution

The equation y^2 = 4ax represents a family of parabolas that open to the right with varying values of 'a'.

Correct Answer: A — Parabolas opening to the right

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Q. What is the length of the diagonal of a rectangle with vertices at (0, 0), (0, 4), (3, 0), and (3, 4)?

A. 3
B. 4
C. 5
D. 6

Solution

Diagonal length = √[(3-0)² + (4-0)²] = √[9 + 16] = √25 = 5.

Correct Answer: C — 5

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Q. What is the length of the diameter of a circle with radius 7?

A. 7
B. 14
C. 21
D. 28

Solution

The diameter is twice the radius, so 2 * 7 = 14.

Correct Answer: B — 14

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Q. What is the length of the diameter of a circle with the equation (x - 1)² + (y + 2)² = 16?

A. 4
B. 8
C. 16
D. 2

Solution

The radius is √16 = 4, so the diameter is 2 * radius = 8.

Correct Answer: B — 8

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Q. What is the length of the latus rectum of the ellipse x^2/36 + y^2/16 = 1?

A. 8/3
B. 12
C. 16/3
D. 24

Solution

The length of the latus rectum is given by (2b^2/a). Here, b^2 = 16, a^2 = 36, so length = (2*16/6) = 12.

Correct Answer: B — 12

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Q. What is the length of the latus rectum of the ellipse x^2/a^2 + y^2/b^2 = 1?

A. 2b^2/a
B. 2a^2/b
C. 2a
D. 2b

Solution

The length of the latus rectum of the ellipse is given by 2a^2/b.

Correct Answer: B — 2a^2/b

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Q. What is the length of the latus rectum of the parabola y^2 = 12x?

A. 3
B. 6
C. 12
D. 24

Solution

The length of the latus rectum of a parabola given by the equation y^2 = 4px is 4p. Here, 4p = 12, so p = 3. Therefore, the length of the latus rectum is 4p = 12.

Correct Answer: B — 6

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Q. What is the length of the latus rectum of the parabola y^2 = 4ax?

A. 2a
B. 4a
C. a
D. None of the above

Solution

The length of the latus rectum of the parabola y^2 = 4ax is 4a.

Correct Answer: A — 2a

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Q. What is the length of the latus rectum of the parabola y^2 = 8x?

A. 4
B. 8
C. 2
D. 16

Solution

The length of the latus rectum of the parabola y^2 = 8x is 4.

Correct Answer: A — 4

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Q. What is the length of the line segment joining the points (-1, -1) and (2, 3)?

A. 3
B. 4
C. 5
D. 6

Solution

Length = √[(2 - (-1))² + (3 - (-1))²] = √[3² + 4²] = √25 = 5.

Correct Answer: C — 5

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Q. What is the length of the line segment joining the points (1, 2) and (1, 5)?

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

Solution

Length = |5 - 2| = 3.

Correct Answer: A — 3

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Q. What is the length of the segment of the line 3x + 4y = 12 between the x-axis and y-axis?

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

Solution

The x-intercept is (4, 0) and the y-intercept is (0, 3). The length of the segment is sqrt((4-0)^2 + (0-3)^2) = sqrt(16 + 9) = 5.

Correct Answer: B — 6

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Q. What is the midpoint of the line segment joining the points (1, 2) and (3, 4)?

A. (2, 3)
B. (1, 2)
C. (3, 4)
D. (4, 6)

Solution

Midpoint = ((1+3)/2, (2+4)/2) = (2, 3).

Correct Answer: A — (2, 3)

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Q. What is the midpoint of the line segment joining the points (2, 3) and (4, 7)?

A. (3, 5)
B. (2, 5)
C. (4, 3)
D. (5, 7)

Solution

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

Correct Answer: A — (3, 5)

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Q. What is the radius of the circle given by the equation (x - 1)² + (y + 4)² = 16?

A. 4
B. 8
C. 2
D. 16

Solution

The radius is the square root of 16, which is 4.

Correct Answer: A — 4

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Q. What is the slope of the line passing through the points (1, 2) and (3, 6)?

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

Solution

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

Correct Answer: A — 2

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

A. 2
B. 1
C. 0.5
D. 3

Solution

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

Correct Answer: A — 2

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Q. What is the slope of the line perpendicular to the line y = -2x + 3?

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

Solution

The slope of the given line is -2. The slope of the perpendicular line is the negative reciprocal, which is 1/2.

Correct Answer: B — 1/2

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Q. What is the slope of the line perpendicular to the line y = -2x + 4?

A. 0.5
B. 2
C. -0.5
D. -2

Solution

The slope of the given line is -2. The slope of the perpendicular line is the negative reciprocal, which is 1/2.

Correct Answer: B — 2

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Q. What is the slope of the line perpendicular to the line y = -3x + 4?

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

Solution

The slope of the given line is -3. The slope of the perpendicular line is the negative reciprocal: 1/3.

Correct Answer: C — 3

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Q. What is the slope of the line represented by the equation 2x - 3y + 6 = 0?

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

Solution

Rearranging gives y = (2/3)x + 2, slope = 2/3.

Correct Answer: C — -2/3

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Q. What is the slope of the line represented by the equation 3x - 4y + 12 = 0?

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

Solution

Rearranging gives y = (3/4)x + 3. Slope = 3/4.

Correct Answer: C — -3/4

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Q. What is the slope of the line represented by the equation 5y - 10x = 20?

A. 2
B. 1
C. 0.5
D. 5

Solution

Rearranging to slope-intercept form: 5y = 10x + 20 => y = 2x + 4. The slope is 2.

Correct Answer: C — 0.5

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Q. What is the slope of the line that passes through the points (0, 0) and (4, 8)?

A. 0
B. 1
C. 2
D. 4

Solution

The slope m = (8 - 0) / (4 - 0) = 2.

Correct Answer: C — 2

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Q. What is the slope of the lines represented by the equation 5x^2 - 10xy + 5y^2 = 0?

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

Solution

The equation can be factored to find the slopes, which are both 1.

Correct Answer: A — 1

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Q. What is the slope of the lines represented by the equation x^2 - 6xy + 9y^2 = 0?

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

Solution

Factoring gives (x - 3y)^2 = 0, indicating a double root, hence the slope is 3.

Correct Answer: A — 3

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Coordinate Geometry

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