Welcome to Part 1 of our 8-part series on Chapter 7: Coordinate Geometry. This post contains the top 25 Multiple Choice Questions (MCQs), focusing on Distance Formula, Section Formula, and Mid-point Formula, with a heavy emphasis on previous year board questions.
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Top 25 MCQs - Coordinate Geometry
Question 1: Distance Formula
The distance between the points (0, 5) and (-5, 0) is:
= √[(-5 - 0)² + (0 - 5)²]
= √[(-5)² + (-5)²] = √[25 + 25] = √50 = 5√2.
Question 2: Distance from Origin
The distance of the point P(2, 3) from the x-axis is:
Question 3: Mid-Point
The mid-point of the line segment joining the points (-5, 7) and (-1, 3) is:
= [(-5 + -1)/2 , (7 + 3)/2]
= [-6/2 , 10/2] = (-3, 5).
Question 4: Value of y
The distance between the points A(0, 6) and B(0, -2) is:
Distance = |-2 - 6| = |-8| = 8.
Question 5: Ratio Division
The point which divides the line segment joining the points (7, -6) and (3, 4) in ratio 1:2 internally lies in the:
x = (1*3 + 2*7)/(1+2) = (3+14)/3 = 17/3 (Positive)
y = (1*4 + 2*(-6))/(1+2) = (4-12)/3 = -8/3 (Negative)
Since x is positive and y is negative, the point lies in the IV quadrant.
Question 6: Origin Distance
The distance of point P(-6, 8) from the origin is:
= √((-6)² + 8²) = √(36 + 64) = √100 = 10.
Question 7: Value of a
If the point P(k, 0) divides the line segment joining the points A(2, -2) and B(-7, 4) in the ratio 1:2, then the value of k is:
k = (1*(-7) + 2*2) / (1+2)
k = (-7 + 4) / 3 = -3 / 3 = -1.
Question 8: Collinear Points
If the points (1, x), (5, 2) and (9, 5) are collinear, then the value of x is:
Slope(AB) = Slope(BC)
(2-x)/(5-1) = (5-2)/(9-5)
(2-x)/4 = 3/4
2-x = 3 => x = -1.
Question 9: Circle Center
The endpoints of a diameter of a circle are (-4, 2) and (8, 6). The coordinates of the center are:
x = (-4 + 8)/2 = 4/2 = 2
y = (2 + 6)/2 = 8/2 = 4
Center is (2, 4).
Question 10: Centroid
The centroid of the triangle whose vertices are (3, -7), (-8, 6) and (5, 10) is:
x = (3 - 8 + 5)/3 = 0/3 = 0
y = (-7 + 6 + 10)/3 = 9/3 = 3
Centroid is (0, 3).
Question 11: X-axis Ratio
The ratio in which the x-axis divides the segment joining (3, 6) and (12, -3) is:
= -(6) : (-3) = -6 : -3 = 2 : 1.
Question 12: Equidistant Point
A point on the y-axis which is equidistant from the points A(6, 5) and B(-4, 3) is:
(0-6)² + (y-5)² = (0-(-4))² + (y-3)²
36 + y² - 10y + 25 = 16 + y² - 6y + 9
61 - 10y = 25 - 6y
36 = 4y => y = 9.
Question 13: Fourth Vertex
Three vertices of a parallelogram are A(1, 2), B(2, 4), and C(5, 9). The coordinates of the fourth vertex D are:
x: (1+5)/2 = (2+x)/2 => 6 = 2+x => x=4
y: (2+9)/2 = (4+y)/2 => 11 = 4+y => y=7
D is (4, 7).
Question 14: Distance a, b
The distance between the points (a cos θ + b sin θ, 0) and (0, a sin θ - b cos θ) is:
Question 15: Perimeter
The perimeter of a triangle with vertices (0, 4), (0, 0) and (3, 0) is:
Side 1 (y-axis) = 4 units. Side 2 (x-axis) = 3 units.
Hypotenuse = √(3² + 4²) = √25 = 5 units.
Perimeter = 3 + 4 + 5 = 12.
Question 16: Y-axis Ratio
The ratio in which the y-axis divides the line segment joining the points (5, -6) and (-1, -4) is:
= -(5) : (-1) = -5 : -1 = 5 : 1.
Question 17: Circle Radius
If the center of a circle is (2a, a-7) and it passes through the point (11, -9). The diameter of the circle is 10√2 units. Find the value of a.
Distance between center and point = radius.
√[(11-2a)² + (-9-(a-7))²] = 5√2
Square both sides: (11-2a)² + (-2-a)² = 50
Solving this quadratic gives a = 5 or a = 3.
Question 18: Area of Rhombus
The vertices of a rhombus are (3, 0), (4, 5), (-1, 4) and (-2, -1). Its area is:
d₁ (distance between 3,0 and -1,4) = √[(-4)² + 4²] = √32 = 4√2.
d₂ (distance between 4,5 and -2,-1) = √[(-6)² + (-6)²] = √72 = 6√2.
Area = 1/2 × 4√2 × 6√2 = 1/2 × 24 × 2 = 24.
Question 19: Trisection
The coordinates of the point which trisects the line segment joining (1, -2) and (-3, 4) near to (1, -2) are:
x = (1*(-3) + 2*1)/3 = -1/3
y = (1*4 + 2*(-2))/3 = 0/3 = 0
Point is (-1/3, 0).
Question 20: Collinear Condition
The points (1, 2), (0, 0) and (a, b) are collinear if:
Slope 1 = (0-2)/(0-1) = 2.
Slope 2 = (b-0)/(a-0) = b/a.
2 = b/a => b = 2a.
Question 21: Third Vertex
If (3, -4) and (-6, 5) are the ends of the diagonal of a parallelogram and (-2, 1) is the third vertex, then the fourth vertex is:
Midpoint (3, -4) & (-6, 5) = (-1.5, 0.5)
Midpoint (-2, 1) & (x, y) = ((-2+x)/2, (1+y)/2)
Solving: -2+x = -3 => x=-1; 1+y = 1 => y=0.
Question 22: Square Vertices
The points A(-1, -2), B(1, 0), C(-1, 2), and D(-3, 0) form a:
Question 23: X-axis Point
A point on x-axis which is equidistant from (2, -5) and (-2, 9) is:
x² - 4x + 29 = x² + 4x + 85.
-8x = 56 => x = -7.
Question 24: Triangle Vertices
The points (3, 2), (-2, -3) and (2, 3) form:
Question 25: Distance Formula
Distance of point (a, b) from (-a, -b) is:
