Class 10 Maths Chapter 7 Coordinate Geometry PYQ

Q1. The distance between the points (2, −2) and (−1, x) is 5. One of the values of x is: (a) −6 (b) 2 (c) −2 (d) 1CBSE 2024 | 1M

Ans: (b) 2. √[(−1−2)² + (x+2)²] = 5 ⇒ 9 + (x+2)² = 25 ⇒ (x+2)² = 16 ⇒ x = 2 or x = −6 [CBSE 2023 | 1 Mark]

Q2. The distance of the point (−3, 4) from the x-axis is: (a) 3 (b) −3 (c) 4 (d) 5CBSE 2023 | 1M

Ans: (c) 4. Distance from x-axis = |y-coordinate| = |4| = 4 units. [CBSE 2024 | 1 Mark]

Q3. The midpoint of the line segment joining A(2, −5) and B(−2, 5) is: (a) (0, 0) (b) (2, −5) (c) (0, 5) (d) (2, 0)CBSE 2024 | 1M

Ans: (a) (0, 0). Midpoint = ((2−2)/2, (−5+5)/2) = (0, 0) UNIQUE STUDY POINT | Amitesh Nagar, Indore (M.P.) | www.uniquestudyonline.com Amitesh Nagar, Indore (M.P.) [CBSE 2023 | 1 Mark]

Q4. If P(a/3, 4) is the midpoint of the line segment joining A(−6, 5) and B(−2, 3), then the value of a is: (a) −4 (b) −12 (c) 12 (d) −6CBSE 2023 | 1M

Ans: (b) −12. a/3 = (−6+(−2))/2 = −4 ⇒ a = −12 [CBSE 2022 | 1 Mark]

Q5. The point on the x-axis which is equidistant from (2, −5) and (−2, 9) is: (a) (−7, 0) (b) (7, 0) (c) (−4, 0) (d) (4, 0)CBSE 2022 | 1M

Ans: (a) (−7, 0). Let P(x, 0). PA² = PB² ⇒ (x−2)²+25 = (x+2)²+81 ⇒ −8x = 56 ⇒ x = −7 [CBSE 2021 | 1 Mark]

Q6. If the point P(k, 0) divides the line segment joining A(2, −2) and B(−7, 4) in the ratio 1:2, then k = (a) 1 (b) 2 (c) −2 (d) −1CBSE 2021 | 1M

Ans: (d) −1. k = (1×(−7) + 2×2)/(1+2) = (−7+4)/3 = −1 [CBSE 2020 | 1 Mark]

Q7. The distance between (a, b) and (−a, −b) is: (a) 2√(a²+b²) (b) √(a²+b²) (c) 2(a+b) (d) 0CBSE 2020 | 1M

Ans: (a) 2√(a²+b²). d = √[(2a)²+(2b)²] = 2√(a²+b²) [CBSE 2022 | 1 Mark]

Q8. If the vertices of a parallelogram ABCD are A(6, 1), B(8, 2), C(9, 4) and D(p, 3), then p = (a) 5 (b) 6 (c) 7 (d) 8CBSE 2022 | 1M

Ans: (c) 7. Diagonals bisect each other. Mid AC = Mid BD. (6+9)/2 = (8+p)/2 ⇒ 15 = 8+p ⇒ p = 7 UNIQUE STUDY POINT | Amitesh Nagar, Indore (M.P.) | www.uniquestudyonline.com Amitesh Nagar, Indore (M.P.) [CBSE 2019 | 1 Mark]

Q9. The coordinates of the point which divides the line joining (1, −2) and (4, 7) in the ratio 1:2 are: (a) (2, 1) (b) (3, 4) (c) (1, 0) (d) (2, 3)CBSE 2019 | 1M

Ans: (a) (2, 1). x = (1×4+2×1)/3 = 6/3 = 2, y = (1×7+2×(−2))/3 = 3/3 = 1 [CBSE 2019 | 1 Mark]

Q10. The distance of the point (3, 5) from the y-axis is: (a) 3 (b) 5 (c) 8 (d) √34CBSE 2019 | 1M

Ans: (a) 3. Distance from y-axis = |x-coordinate| = |3| = 3 units. [CBSE 2024 | 1 Mark]

Q11. Assertion (A): The point (−1, 6) divides the line segment joining A(−3, 10) and B(6, −8) in the ratio 2:7. Reason (R): The section formula for internal division is: ((m₁x₂ + m₂x₁)/(m₁+m₂), (m₁y₂ + m₂y₁)/(m₁+m₂)). (a) Both true, R is correct explanation of A (b) Both true, R is not correct explanation (c) A is true, R is false (d) A is false, R is trueCBSE 2024 | 1M

Ans: (a) Both true and R explains A. Using section formula 2:7: x = (2×6+7×(−3))/9 = (12−21)/9 = −1 ✔ [CBSE 2023 | 1 Mark]

Q12. Assertion (A): The midpoint of a line segment divides it in the ratio 1:1. Reason (R): The midpoint formula is ((x₁+x₂)/2, (y₁+y₂)/2). (a) Both true, R is correct explanation of A (b) Both true, R is not correct explanation (c) A is true, R is false (d) A is false, R is trueCBSE 2023 | 1M

Ans: (a) Both true and R explains A. Midpoint divides in 1:1 and the formula is a special case of section formula. [CBSE 2024 | 2 Marks]

Q13. Find the ratio in which the y-axis divides the line segment joining the points (5, 3) and (−1, 6).CBSE 2024 | 2M

Ans: Let ratio = k:1. x = (k(−1)+1(5))/(k+1) = 0 ⇒ −k+5 = 0 ⇒ k = 5. Ratio = 5:1. UNIQUE STUDY POINT | Amitesh Nagar, Indore (M.P.) | www.uniquestudyonline.com Amitesh Nagar, Indore (M.P.) [CBSE 2022 | 2 Marks]

Q14. Find a point on the x-axis which is equidistant from the points (7, 6) and (3, 4).CBSE 2022 | 2M

Ans: Let P(x, 0). PA² = PB² ⇒ (x−7)²+36 = (x−3)²+16 ⇒ x²−14x+49+36 = x²−6x+9+16 ⇒ −8x = −60 ⇒ x = 7.5. Point: (7.5, 0) [CBSE 2021 | 2 Marks]

Q15. Find the coordinates of the points of trisection of the line segment joining (3, −2) and (−3, −4).CBSE 2021 | 2M

Ans: P divides in 1:2: P = (1(−3)+2(3))/3, (1(−4)+2(−2))/3) = (1, −8/3). Q divides in 2:1: Q = (2(−3)+1(3))/3, (2(−4)+1(−2))/3) = (−1, −10/3). [CBSE 2020 | 2 Marks]

Q16. Show that the points A(1, 7), B(4, 2), C(−1, −1) and D(−4, 4) are the vertices of a square.CBSE 2020 | 2M

Ans: AB = √(9+25) = √34. BC = √(25+9) = √34. CD = √(9+25) = √34. DA = √(25+9) = √34. All sides equal. AC = √(4+64) = √68. BD = √(64+4) = √68. Diagonals equal. Hence ABCD is a square. [CBSE 2024 | 3 Marks]

Q17. Points A(−1, y) and B(5, 7) lie on a circle with centre O(2, −3y) such that AB is a diameter. Find y and the radius.CBSE 2024 | 3M

Ans: O is midpoint of AB: 2 = (−1+5)/2 ✔. −3y = (y+7)/2 ⇒ −6y = y+7 ⇒ −7y = 7 ⇒ y = −1. O = (2, 3), A = (−1, −1). Radius = OA = √(9+16) = 5 units. [CBSE 2022 | 3 Marks]

Q18. If A(1, 2), B(4, 3) and C(6, 6) are the three vertices of a parallelogram ABCD, find the coordinates of the fourth vertex D.CBSE 2022 | 3M

Ans: Diagonals bisect: Mid AC = Mid BD. Mid AC = ((1+6)/2, (2+6)/2) = (3.5, 4). Mid BD: (3.5, 4) = ((4+x)/2, (3+y)/2) ⇒ x = 3, y = 5. D = (3, 5). [CBSE 2021 | 3 Marks]

Q19. In what ratio does the point P(2, 5) divide the line segment joining A(8, 2) and B(−6, 9)?CBSE 2021 | 3M

Ans: Let ratio = k:1. x: (k(−6)+8)/(k+1) = 2 ⇒ −6k+8 = 2k+2 ⇒ 8k = 6 ⇒ k = 3/4. Ratio = 3:4. [CBSE 2019 | 3 Marks]

Q20. Find the coordinates of the point which divides the line segment joining (−1, 7) and (4, −3) in the ratio 2:3.CBSE 2019 | 3M

Ans: x = (2(4)+3(−1))/(2+3) = (8−3)/5 = 1. y = (2(−3)+3(7))/(2+3) = (−6+21)/5 = 3. Point = (1, 3). UNIQUE STUDY POINT | Amitesh Nagar, Indore (M.P.) | www.uniquestudyonline.com Amitesh Nagar, Indore (M.P.) [CBSE 2023 | 5 Marks]

Q21. If A(−5, 7), B(−4, −5), C(−1, −6) and D(4, 5) are the vertices of a quadrilateral, find its type by calculating lengths of all sides and diagonals.CBSE 2023 | 5M

Ans: AB = √(1+144) = √145. BC = √(9+1) = √10. CD = √(25+121) = √146. DA = √(81+4) = √85. Since AB ≠ CD and BC ≠ DA, it is neither a parallelogram nor any special quad. It is a general quadrilateral. [CBSE 2020 | 5 Marks]

Q22. The coordinates of two adjacent vertices of a parallelogram are (3, 2) and (−1, 0), and its diagonals intersect at (2, −5). Find the coordinates of the other two vertices.CBSE 2020 | 5M

Ans: Let A(3,2), B(−1,0). Diagonal intersection O(2,−5). Mid AC = O: C = (2×2−3, 2×(−5)−2) = (1, −12). Mid BD = O: D = (2×2−(−1), 2×(−5)−0) = (5, −10). Other vertices: C(1, −12) and D(5, −10). [CBSE 2025 | 4 Marks]

Q23. Case Study: In a GPS-based city map, a school is located at A(3, 4), a hospital at B(7, 4) and a park at C(5, 8). (i) Find the distance between the school and hospital. (ii) Find the midpoint of the line segment AB. (iii) If a library D divides AC in ratio 1:1, find its coordinates. (iv) Find the distance of the park from the x-axis.CBSE 2025 | 4M

Ans: (i) AB = √[(7−3)²+0] = 4 units. (ii) Mid AB = ((3+7)/2, (4+4)/2) = (5, 4). (iii) D = midpoint of AC = ((3+5)/2, (4+8)/2) = (4, 6). (iv) Distance from x-axis = y-coordinate of C = 8 units. [CBSE 2024 | 4 Marks]

Q24. Case Study: Two friends Raj and Ajay are standing at points P(2, 8) and Q(6, 4) in a park. Their friend Simran wants to stand exactly between them. (i) Find the coordinates where Simran should stand. (ii) Find the distance between Raj and Ajay. (iii) If another friend stands at R(10, 0), find the distance QR. (iv) Find the point that divides PQ in ratio 3:1.CBSE 2024 | 4M

Ans: (i) Midpoint PQ = ((2+6)/2, (8+4)/2) = (4, 6). (ii) PQ = √(16+16) = 4√2 units. (iii) QR = √(16+16) = 4√2 units. (iv) x = (3(6)+1(2))/4 = 5, y = (3(4)+1(8))/4 = 5. Point = (5, 5). UNIQUE STUDY POINT | Amitesh Nagar, Indore (M.P.) | www.uniquestudyonline.com Amitesh Nagar, Indore (M.P.) ★ PYQ SUMMARY & ANALYSIS Topic Years Asked Frequency Marks Distance Formula 2019–2025 Every Year 1–3 Section Formula (ratio) 2019–2025 Every Year 1–3 Midpoint Formula 2019–2025 Every Year 1–2 Equidistant point problems 2019–2024 5 times 2–3 Parallelogram vertices (diag. bisect) 2019–2024 4 times 3–5 Trisection of line segment 2019–2023 3 times 2–3 Prove quadrilateral type 2019–2023 3 times 3–5 Case Study (map/coordinate) 2024–2025 2 times 4 Key Observations for Students: ✔ Distance Formula: d = √[(x₂−x₁)² + (y₂−y₁)²] — used in 90% of problems. ✔ Section Formula: P = ((m₁x₂+m₂x₁)/(m₁+m₂), (m₁y₂+m₂y₁)/(m₁+m₂)) ✔ Midpoint is special case of Section Formula with ratio 1:1. ✔ "Equidistant from" ⇒ PA = PB ⇒ PA² = PB² (square both sides, cancel x²). ✔ Parallelogram: diagonals bisect each other — very frequent 3–5 mark question. ✘ Area of Triangle formula is DELETED from 2025–26 syllabus. ✔ Expected marks: 6–8 marks in Board Exam. "Practice makes perfect. Solve PYQs to master your Board Exam!" Best Wishes for Your Board Exam! Visit: www.uniquestudyonline.com Download Our App: Search "Unique Study Point" on Google Play Store UNIQUE STUDY POINT | Amitesh Nagar, Indore (M.P.) | www.uniquestudyonline.com