35 Solved Coordinate Geometry Questions for Class 10 Math

35 Solved Coordinate Geometry Questions

1. Find the distance between (2, 3) and (4, 1).

$d = \sqrt{(4-2)^2 + (1-3)^2} = \sqrt{2^2 + (-2)^2} = \sqrt{8} = $ 2√2 units

2. Find the distance between (-5, 7) and (-1, 3).

$d = \sqrt{(-1 - (-5))^2 + (3-7)^2} = \sqrt{4^2 + (-4)^2} = \sqrt{32} = $ 4√2 units

3. Distance of point (x, y) from origin (0, 0)?

Distance = $\sqrt{x^2 + y^2}$

4. Find distance between (a, b) and (-a, -b).

$d = \sqrt{(-a-a)^2 + (-b-b)^2} = \sqrt{(-2a)^2 + (-2b)^2} = \sqrt{4a^2 + 4b^2} = $ 2√(a² + b²)

5. Are points (1, 5), (2, 3) and (-2, -11) collinear?

Check if Area of Triangle = 0. $1(3 - (-11)) + 2(-11 - 5) + (-2)(5 - 3) = 14 - 32 - 4 = -22 \neq 0$.
Result: Not Collinear.

6. Find midpoint of (6, 8) and (2, 4).

$M = (\frac{6+2}{2}, \frac{8+4}{2}) = (4, 6)$.

7. Find 'y' if distance between (2, -3) and (10, y) is 10.

$10^2 = (10-2)^2 + (y+3)^2 \Rightarrow 100 = 64 + (y+3)^2 \Rightarrow 36 = (y+3)^2$.
$y+3 = \pm 6 \Rightarrow y = 3$ or $y = -9$.

8. Relation between x and y such that (x, y) is equidistant from (7, 1) and (3, 5)?

$(x-7)^2 + (y-1)^2 = (x-3)^2 + (y-5)^2$. Simplifying: x - y = 2.

9. Point on x-axis equidistant from (2, -5) and (-2, 9).

Point is (x, 0). $(x-2)^2 + (0+5)^2 = (x+2)^2 + (0-9)^2 \Rightarrow -8x = 56 \Rightarrow x = -7$.
Point: (-7, 0).

10. Find the centroid of triangle with vertices (1,1), (0,0), and (2,2).

$G = (\frac{1+0+2}{3}, \frac{1+0+2}{3}) = (1, 1)$.

11. Find coordinates dividing (-1, 7) and (4, -3) in ratio 2:3.

$x = \frac{2(4)+3(-1)}{5} = 1, y = \frac{2(-3)+3(7)}{5} = 3$. Point: (1, 3).

12. Coordinates of points of trisection of (4, -1) and (-2, -3).

Ratio 1:2 $\Rightarrow$ (2, -5/3); Ratio 2:1 $\Rightarrow$ (0, -7/3).

13. Find ratio in which y-axis divides (-4, 5) and (3, -7).

On y-axis, x=0. $0 = \frac{k(3) + 1(-4)}{k+1} \Rightarrow 3k = 4 \Rightarrow k = 4:3$.

14. Ratio in which (-3, 10) and (6, -8) is divided by (-1, 6).

$-1 = \frac{6k - 3}{k+1} \Rightarrow -k-1 = 6k-3 \Rightarrow 7k=2 \Rightarrow$ 2:7.

15. Area of triangle with vertices (2, 3), (-1, 0), (2, -4).

$\frac{1}{2}|2(0 - (-4)) + (-1)(-4 - 3) + 2(3 - 0)| = \frac{1}{2}|8 + 7 + 6| = $ 10.5 sq. units.

16. Find 'k' if (2, 3), (4, k), (6, -3) are collinear.

Area = 0 $\Rightarrow 2(k+3) + 4(-3-3) + 6(3-k) = 0 \Rightarrow -4k = 0 \Rightarrow k = 0$.

17. Perimeter of triangle with vertices (0,4), (0,0), (3,0).

Sides: 4, 3, and $\sqrt{3^2+4^2}=5$. Perimeter = $4+3+5 = 12$ units.

18. Find fourth vertex of parallelogram: (1,2), (4,3), (6,6).

Midpoints of diagonals are same. $(1+6)/2 = (4+x)/2 \Rightarrow x=3$. $(2+6)/2 = (3+y)/2 \Rightarrow y=5$. Vertex: (3, 5).

19. Coordinate of point A where AB is diameter, center is (2, -3), B is (1, 4).

$2 = (x+1)/2 \Rightarrow x=3$. $-3 = (y+4)/2 \Rightarrow y=-10$. A is (3, -10).

20. Distance between $(a\cos\theta, 0)$ and $(0, a\sin\theta)$.

$\sqrt{(0-a\cos\theta)^2 + (a\sin\theta-0)^2} = \sqrt{a^2(\cos^2\theta + \sin^2\theta)} = \sqrt{a^2} = a$.

21. If P(x,y) is equidistant from A(5,1) and B(-1,5), find relation.

$(x-5)^2 + (y-1)^2 = (x+1)^2 + (y-5)^2 \Rightarrow 3x = 2y$.

22. Find area of rhombus if vertices are (3,0), (4,5), (-1,4), (-2,-1).

$\frac{1}{2} \times d_1 \times d_2$. $d_1 = \sqrt{32}, d_2 = \sqrt{72}$. Area = 24 sq. units.

23. Point P divides AB in ratio 1:3. A(2,1), B(7,6). Find P.

$x = \frac{1(7)+3(2)}{4} = 3.25, y = \frac{1(6)+3(1)}{4} = 2.25$.

24. Find distance of (4, -3) from x-axis.

Distance from x-axis = $|y| = |-3| = 3$ units.

25. Find distance of (4, -3) from y-axis.

Distance from y-axis = $|x| = 4$ units.

26. If points (0,0), (3,√3) and (3,p) form equilateral triangle, find p.

Distance OP = PQ. $p = -\sqrt{3}$.

27. Midpoint of line joining (2a, 4) and (-2, 3b) is (1, 2a+1). Find a, b.

$(2a-2)/2 = 1 \Rightarrow a=2$. $(4+3b)/2 = 2(2)+1 \Rightarrow b=2$.

28. Find area of triangle with vertices (a, b+c), (b, c+a), (c, a+b).

$Area = 0$ (Points are collinear).

29. Find distance between $(L, M)$ and $(L+a, M+b)$.

$d = \sqrt{(L+a-L)^2 + (M+b-M)^2} = \sqrt{a^2+b^2}$.

30. Coordinate of point on y-axis equidistant from (6,5) and (-4,3).

Point (0, y). $6^2 + (y-5)^2 = (-4)^2 + (y-3)^2 \Rightarrow y = 9$. Point: (0, 9).

31. Ratio in which line segment joining (1,-5) and (-4,5) is divided by x-axis.

$y=0 \Rightarrow 0 = \frac{k(5) + 1(-5)}{k+1} \Rightarrow k=1$. Ratio: 1:1.

32. Centroid of triangle with vertices (a,b), (b,c), (c,a) is origin. Find a+b+c.

$(a+b+c)/3 = 0 \Rightarrow a+b+c = 0$.

33. Distance between $(\cos\theta, \sin\theta)$ and $(\sin\theta, \cos\theta)$.

$\sqrt{(\sin\theta-\cos\theta)^2 + (\cos\theta-\sin\theta)^2} = \sqrt{2(\sin\theta-\cos\theta)^2} = \sqrt{2}|\sin\theta-\cos\theta|$.

34. Find 'p' if (p, 2) is midpoint of (5, 3) and (1, 1).

$p = (5+1)/2 = 3$.

35. Type of triangle formed by (1,1), (-1,-1), (√3, -√3).

$AB = \sqrt{8}, BC = \sqrt{8}, AC = \sqrt{8}$. Result: Equilateral Triangle.