Chapter 7: Coordinate Geometry
CBSE Official PYQs (2015-2026)
Q1. The distance of the point \(P(-6, 8)\) from the origin is :
Q2. In what ratio does the x-axis divide the line segment joining the points \(A(3, 6)\) and \(B(-12, -3)\)?
Q3. The distance between the points \((0, 2\sqrt{5})\) and \((-2\sqrt{5}, 0)\) is :
Q4. \(AOBC\) is a rectangle whose three vertices are \(A(0, 2)\), \(O(0, 0)\) and \(B(4, 0)\). The square of the length of its diagonal is equal to :
Q5. Point \(P\) divides the line segment joining the points \(A(4, -5)\) and \(B(1, 2)\) in the ratio \(5 : 2\). The coordinates of point \(P\) are :
Q6. The distance of the point \(P(-3, 4)\) from the x-axis is :
Q7. The midpoint of the line segment joining the points \(P(-2, 8)\) and \(Q(-6, -4)\) is :
Q8. If the distance between the points \((4, p)\) and \((1, 0)\) is 5 units, then the value of \(p\) is :
Q9. The perimeter of a triangle with vertices \((0, 4)\), \((0, 0)\) and \((3, 0)\) is :
Q10. The point on the x-axis which is equidistant from the points \(A(-1, 0)\) and \(B(5, 0)\) is :
Q11. 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 :
Q12. The distance of the point \(P(5, -12)\) from the origin is :
Q13. The ratio in which the y-axis divides the line segment joining the points \(A(5, -6)\) and \(B(-1, -4)\) is :
Q14. The vertices of a triangle \(ABC\) are plotted on a coordinate grid at points \(A(1, 2)\), \(B(5, 2)\) and \(C(3, 5)\). What are the coordinates of the centroid of triangle \(ABC\)?
Q15. If the coordinates of one end of a diameter of a circle are \((2, 3)\) and the coordinates of its centre are \((-2, 5)\), then the coordinates of the other end of the diameter are :
Q16. The point \(P\) which divides the line segment joining the points \(A(2, -5)\) and \(B(5, 2)\) in the ratio \(2 : 3\) lies in which quadrant?
Q17. If the distance between the points \(A(x, 2)\) and \(B(3, -6)\) is 10 units, then the positive value of \(x\) is :
Q18. The line segment joining the points \(A(-2, 9)\) and \(B(6, 3)\) is a diameter of a circle. The coordinates of the centre of the circle are :
Q19. If the coordinates of the midpoint of the line segment joining \(A(3, 4)\) and \(B(k, 6)\) are \((2, 5)\), then the value of \(k\) is :
Q20. If \(P(a/3, 4)\) is the midpoint of the line segment joining the points \(Q(-6, 5)\) and \(R(-2, 3)\), then the value of '\(a\)' is :
Q21. The distance of the point \(P(-5, 12)\) from the y-axis is :
Q22. The coordinates of the point which divides the line segment joining the points \((1, 3)\) and \((2, 7)\) in the ratio \(3 : 4\) internally are :
Q23. If the mid-point of the segment joining \(A(x, y+1)\) and \(B(x+1, y+2)\) is \((1.5, 2.5)\), then the values of \(x\) and \(y\) are respectively :
Q24. Three points are plotted with coordinates \(A(1, 1)\), \(B(3, 1)\) and \(C(4, 2)\). Find the coordinates of the point \(D\) such that \(ABCD\) forms a parallelogram.
Q25. If the centroid of the triangle formed by the points \((a, b)\), \((b, c)\) and \((c, a)\) is at the origin, then \(a + b + c\) is equal to :
Q26. If the distance between the points \((4, k)\) and \((1, 0)\) is 5, then what is/are the possible value(s) of \(k\)?
Q27. If the points \(A(1, 2)\), \(O(0, 0)\) and \(C(a, b)\) are collinear, then which of the following relations is correct?
Q28. If the line segment joining \(A(2, 3)\) and \(B(-3, 8)\) is divided by a point \(P\) on the y-axis, then the y-coordinate of point \(P\) is :
Q29. The distance of the point \(P(-2, -3)\) from the x-axis is :
Q30. The ratio in which the point \(P(4, 6)\) divides the line segment joining the points \(A(2, 3)\) and \(B(6, 9)\) is :
Q31. Find a relation between \(x\) and \(y\) such that the point \(P(x, y)\) is equidistant from the points \(A(7, 1)\) and \(B(3, 5)\).
Q32. Find the point on the x-axis which is equidistant from the points \((2, -5)\) and \((-2, 9)\).
Q33. Find the coordinates of the points of trisection of the line segment joining the points \(A(2, -2)\) and \(B(-7, 4)\).
Q34. Find the ratio in which the y-axis divides the line segment joining the points \((5, -6)\) and \((-1, -4)\). Also, find the coordinates of the point of division.
Q35. If the point \(C(-1, 2)\) divides internally the line segment joining the points \(A(2, 5)\) and \(B(x, y)\) in the ratio \(3 : 4\), find the coordinates of \(B\).
Q36. If the points \(A(6, 1)\), \(B(8, 2)\), \(C(9, 4)\) and \(D(p, 3)\) are the vertices of a parallelogram, taken in order, find the value of \(p\).
Q37. Find the coordinates of a point \(A\), where \(AB\) is the diameter of a circle whose centre is \((2, -3)\) and \(B\) is \((1, 4)\).
Q38. Find the distance between the points \(A(a \cos\theta + b \sin\theta, 0)\) and \(B(0, a \sin\theta - b \cos\theta)\).
Q39. Find a relation between \(x\) and \(y\) such that the point \(P(x, y)\) is equidistant from the points \(A(3, 6)\) and \(B(-3, 4)\).
Q40. Find the ratio in which the line segment joining the points \(A(1, -5)\) and \(B(-4, 5)\) is divided by the x-axis.
Q41. Name the type of quadrilateral formed, if any, by the points \(A(-1, -2)\), \(B(1, 0)\), \(C(-1, 2)\) and \(D(-3, 0)\), and give reasons for your answer.
Q42. Show that the points \(A(3, 0)\), \(B(6, 4)\) and \(C(-1, 3)\) are the vertices of an isosceles right-angled triangle.
Q43. If the point \(P(x, y)\) is equidistant from the points \(A(a + b, b - a)\) and \(B(a - b, a + b)\), then prove that \(bx = ay\).
Q44. The line segment joining the points \(A(2, 1)\) and \(B(5, -8)\) is trisected at the points \(P\) and \(Q\) such that \(P\) is nearer to \(A\). If \(P\) also lies on the line given by \(2x - y + k = 0\), find the value of \(k\).
Q45. If the vertices of a triangle are \(A(1, -1)\), \(B(0, 4)\) and \(C(-5, 3)\), find the length of the median \(AD\) drawn from vertex \(A\).
Q46. Find the coordinates of the points which divide the line segment joining \(A(-2, 2)\) and \(B(2, 8)\) into four equal parts.
Q47. If the point \(P(x, y)\) is equidistant from the points \(A(5, 1)\) and \(B(-1, 5)\), prove that \(3x = 2y\).
Q48. Find the values of \(y\) for which the distance between the points \(P(2, -3)\) and \(Q(10, y)\) is 10 units.
Q49. Case Study: Sports Day Activities
To conduct Sports Day activities in a rectangular shaped school ground \(ABCD\), lines have been drawn with chalk powder at a distance of 1 m each. 100 flower pots have been placed at a distance of 1 m from each other along \(AD\).
Niharika runs \(\frac{1}{4}\text{th}\) the distance \(AD\) on the 2nd line and posts a green flag. Preet runs \(\frac{1}{5}\text{th}\) the distance \(AD\) on the 8th line and posts a red flag.
Based on the above information, answer the following questions:
- What is the distance between both the flags? (2 Marks)
- If Rashmi has to post a blue flag exactly halfway between the line segment joining the two flags, where should she post her flag? (2 Marks)
- If Joy has to post a yellow flag at a distance of \(\frac{1}{3}\text{rd}\) from the green flag on the line segment joining them, find its coordinates. (1 Mark)
1. \(\sqrt{61}\text{ m}\) (Green flag at \((2, 25)\) and Red flag at \((8, 20)\))
2. 5th line at a distance of \(22.5\text{ m}\) (Coordinates: \((5, 22.5)\))
3. \((4, 23.33)\)
Q50. Case Study: GPS Tracking on a Golf Course
During a golf tournament, a GPS tracking system records the positions of three golf balls on a grid layout of the course. The positions are recorded as:
- Ball A at \((2, 3)\)
- Ball B at \((6, 7)\)
- Ball C at \((10, 11)\)
Based on the above coordinates, answer the following questions:
- Find the distance between Ball A and Ball B. (2 Marks)
- Determine if the positions of the three golf balls A, B, and C are collinear. Show your calculation. (2 Marks)
- If a target hole is located at the coordinates \((14, 15)\), is it on the same straight line trajectory as the three balls? (1 Mark)
1. \(4\sqrt{2}\text{ units}\)
2. Yes, they are collinear (\(AB + BC = AC\))
3. Yes, it satisfies the linear trajectory.
Q51. Case Study: Seating Arrangement in a Seminar Hall
In a seminar hall, seats are arranged in a grid-like coordinate system. The positions of three friends sitting at points A, B, and C are plotted on a cartesian layout:
- Ananya (A) is seated at \((3, 1)\)
- Bhuvan (B) is seated at \((6, 4)\)
- Chaitanya (C) is seated at \((8, 6)\)
Based on this seating coordinates:
- Find the distance between Ananya (A) and Bhuvan (B). (2 Marks)
- If another student, Deepak, sits at a point \(D(x, y)\) such that \(ABCD\) forms a parallelogram in that order, find the coordinates of Deepak (D). (2 Marks)
- Find the coordinates of the midpoint of the diagonal \(AC\). (1 Mark)
1. \(3\sqrt{2}\text{ units}\)
2. Deepak (D) is seated at \((5, 3)\)
3. \((5.5, 3.5)\)
Q52. Case Study: Model Town Residential Layout
A developer is designing a residential sector 'Model Town' on a grid sheet. He marks the entry gate at origin \(O(0, 0)\), a community park at \(P(4, 6)\), and a shopping complex at \(S(8, 2)\).
Based on the layout:
- Find the direct distance between the community park P and the shopping complex S. (2 Marks)
- The developer wants to build a central fountain F on the line segment joining the park P and the shopping complex S such that it divides PS in the ratio \(3 : 1\). Find the coordinates of the fountain F. (2 Marks)
- Check whether the entry gate \(O(0, 0)\), park \(P(4, 6)\) and shopping complex \(S(8, 2)\) form a collinear layout. (1 Mark)
1. \(4\sqrt{2}\text{ units}\)
2. \(F(7, 3)\)
3. No, they are not collinear (Area of triangle OPS \(\neq 0\)).