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Math Olympiad Test: Coordinate Geometry- 2 - Class 10 MCQ


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15 Questions MCQ Test Olympiad Preparation for Class 10 - Math Olympiad Test: Coordinate Geometry- 2

Math Olympiad Test: Coordinate Geometry- 2 for Class 10 2024 is part of Olympiad Preparation for Class 10 preparation. The Math Olympiad Test: Coordinate Geometry- 2 questions and answers have been prepared according to the Class 10 exam syllabus.The Math Olympiad Test: Coordinate Geometry- 2 MCQs are made for Class 10 2024 Exam. Find important definitions, questions, notes, meanings, examples, exercises, MCQs and online tests for Math Olympiad Test: Coordinate Geometry- 2 below.
Solutions of Math Olympiad Test: Coordinate Geometry- 2 questions in English are available as part of our Olympiad Preparation for Class 10 for Class 10 & Math Olympiad Test: Coordinate Geometry- 2 solutions in Hindi for Olympiad Preparation for Class 10 course. Download more important topics, notes, lectures and mock test series for Class 10 Exam by signing up for free. Attempt Math Olympiad Test: Coordinate Geometry- 2 | 15 questions in 15 minutes | Mock test for Class 10 preparation | Free important questions MCQ to study Olympiad Preparation for Class 10 for Class 10 Exam | Download free PDF with solutions
Math Olympiad Test: Coordinate Geometry- 2 - Question 1
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Find the third vertex of a ΔABC if two of its vertices are B (-3, 1), C (0, -2), and its centroid is at the origin.

Detailed Solution for Math Olympiad Test: Coordinate Geometry- 2 - Question 1

B  (-3, 1) C (0, -2)
Let the third vertex be A (x, y)
G = (0, 0)
= 0 ⇒ x = 3 and 
⇒ y - 1 = 0 ⇒ y = 1
A (3, 1)

Math Olympiad Test: Coordinate Geometry- 2 - Question 2
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The circumcentre of a triangle is (3, 3). If its two vertices are (4, 6) and (0, 4) find the third vertex of the triangle.

Detailed Solution for Math Olympiad Test: Coordinate Geometry- 2 - Question 2

Let P (3, 3) be the circumcentre of ΔABC, C(x, y) be the third vertex

PA = PB = PC
PA2 = PB2 = PC2
PA2 = PC2
⇒ (4 - 3)2 + (6 - 3)2
= (x - 3)2 + (y - 3)2
⇒ 1 + 9 = x2 + 9 - 6x + y2 + 9 - 6y
⇒ x2 + y2 - 6x - 6y + 8 = 0 ...(1)
and PB2 = PC2
⇒ (0 - 3)2 + (4, -3)2 = (x - 3)2 + (y - 3)2
⇒ 9 + 1 = x2 + 9 - 6x + y2 - 6y + 9
Equations (1) and (2) are identical
(x - 3)2 + (y - 3 )2 = 10
So, (6, 2) is the required point.

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Math Olympiad Test: Coordinate Geometry- 2 - Question 3
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In which ratio does the point P (1, 2) divides the join of A (-2, 1) and B (7, 4)?

Detailed Solution for Math Olympiad Test: Coordinate Geometry- 2 - Question 3


Let the ratio be k : 1

⇒ 6k = 3
⇒ k = 3/6 = 1/2

Math Olympiad Test: Coordinate Geometry- 2 - Question 4
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If (2, -2), (-2, 1) and (5, 2) are vertices of a right angled triangle, then the area of triangle is
Detailed Solution for Math Olympiad Test: Coordinate Geometry- 2 - Question 4

Area of triangle 
= 1/2 [x1 (y2 - y3) + x2 (y3 -y1) + x3 (y1 - y2) 
= 1/2 [2 (1 − 2) + (−2) (2 + 2) + 5 (−2 −1)]
= 1/2 [−2−8 −15] = 25/2 = 12.5 sq. units

Math Olympiad Test: Coordinate Geometry- 2 - Question 5
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The centre of a circle is (4, 5) A(8, 10) is a point on the circumference. Find the other end of diameter of the circle through A.
Detailed Solution for Math Olympiad Test: Coordinate Geometry- 2 - Question 5

Let the other end be (x, y)

Math Olympiad Test: Coordinate Geometry- 2 - Question 6
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The ends of a diameter of a circle have the coordinates (4, 3) and (-4, -3), PQ is another diameter where P has co-ordinate 
 
find the coordinates of Q.
Detailed Solution for Math Olympiad Test: Coordinate Geometry- 2 - Question 6

Mid-point of (4, 3) and (-4, -3)

Math Olympiad Test: Coordinate Geometry- 2 - Question 7
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The vertices of a triangle are A(1, 1), B(-2, -5) and C(2, 2) find the length of the medium through C.
Detailed Solution for Math Olympiad Test: Coordinate Geometry- 2 - Question 7


Math Olympiad Test: Coordinate Geometry- 2 - Question 8
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What is the circumradius of the triangle whose vertices are (2, -2), (8, 6), and (8, -2)?
Detailed Solution for Math Olympiad Test: Coordinate Geometry- 2 - Question 8


Let P(x, y) be the circumcentre
PA = PB = PC
⇒ PA2 = PB2
⇒ (x - 2)2 + (y + 2)2
= (x - 8)2 + (y - 6)2
⇒ x2 + 4 - 4x + y2 + 4 - 4y
= x2+ 64 - 16x + y2 + 36 - 12y
⇒ - 4x + 16x + 4y + 12y = 100 - 8
⇒ 12x + 16y = 92 ⇒ 3x+ 4y = 23 ...(1)
and PB2 = PC2
⇒ (x - 8)2 + (y - 6)2 = (x - 8)2 + (y + 2)2
⇒ y2 + 36 - 12y = y2 + 4 + 4y
⇒ 16y = 32 ⇒ y = 2
3x = 23 - 4 ⇒ 2 = 15 ⇒ x = 5
P (5, 2)
∴ PA

Math Olympiad Test: Coordinate Geometry- 2 - Question 9
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The sum of the square of the distance of a moving point from two fixed points (a, 0) and (-a, 0) is equal to the constant quantity 2c2. Find the equation of its locus.
Detailed Solution for Math Olympiad Test: Coordinate Geometry- 2 - Question 9


Here (x -a)2 + (y - 0)2 = 2c2
x2 + a2 - 2ax + y2 = 2c2
and (x + a)2 + (y - 0)2 = 2c2
⇒ x2 + a2 + 2ax + y2 + 2c2
Now x2 + a2 - 2ax + y2 = x2 + a2 + 2ax + y2
⇒ 4ax = 0 ⇒ x = 0 or a = 0
(x - 0)2 + y2 = 2c2
⇒ x0 + y2 = 2c2

Math Olympiad Test: Coordinate Geometry- 2 - Question 10
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What is the distance of the point (4, 7) from the y-axis?
Detailed Solution for Math Olympiad Test: Coordinate Geometry- 2 - Question 10

Distance of the point (4, 7) from (0, y)
= x co-ordinate
= 4

Math Olympiad Test: Coordinate Geometry- 2 - Question 11
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If the distance between the points (3, 0) and (0, y) is 5 units. y is positive then what is value of y?
Detailed Solution for Math Olympiad Test: Coordinate Geometry- 2 - Question 11

(3 - 0)2 + (0 - y)2 = 52
9 + y2 = 25 ⇒ y2 = 16 ⇒ y = ± 4
y is positive, ∴ y = 4

Math Olympiad Test: Coordinate Geometry- 2 - Question 12
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If the centroid of the triangle is formed by points P(a, b), Q(b, c) and R(c, a) is at the origin. What is the value of a + b +c?
Detailed Solution for Math Olympiad Test: Coordinate Geometry- 2 - Question 12

Math Olympiad Test: Coordinate Geometry- 2 - Question 13
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If the point P(-1, 2) divides externally the line segment joining A(2, 5) and B in the ratio 3 : 4. What is the co-ordinate of point B?
Detailed Solution for Math Olympiad Test: Coordinate Geometry- 2 - Question 13

Math Olympiad Test: Coordinate Geometry- 2 - Question 14
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The points A(3, 1), B(0, 4), C(-3, 1), D(0, -2) are vertices of a
Detailed Solution for Math Olympiad Test: Coordinate Geometry- 2 - Question 14

The given points are A (0,-2), B(3,1), C(0,4) and D(-3,1)

Thus, diagonal PR = diagonal QS 
Therefore, the given points from a square. 

Math Olympiad Test: Coordinate Geometry- 2 - Question 15
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If the point (x, 4) lies on a circle whose center is at the origin and radius is 5, What is the value of x?
Detailed Solution for Math Olympiad Test: Coordinate Geometry- 2 - Question 15


OP = 5
⇒ OP2 = 25
⇒ (x - 0)2 + (4 - 0)2 = 25
⇒ x2 + 16 = 25
⇒ x2 = 9 ⇒ x = ± 3

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