Class 10 Maths · Chapter 7

Coordinate Geometry

440 Class 10 Maths questions on Coordinate Geometry from the NCERT textbook, NCERT Exemplar and CBSE board papers 2015–2026, each with its answer and a step-by-step solution.

CBSE Class 10 Maths previous year papers, with answers →

Questions · page 9 of 22

1. Q161 CBSE 2025
   Using distance formula, prove that ​the ​points $A(3,1)$, $B(6,4)$ and $C(8,6)$ ​are ​collinear.
   Show answer

   $AB=3\sqrt{2}$, $BC=2\sqrt{2}$ and $AC=5\sqrt{2}$. Since $AB+BC=5\sqrt{2}=AC$, the points $A$, $B$ and $C$ are collinear. Hence proved.

   Step-by-step solution →
2. Q162 CBSE 2025
   Show that ​the points $(a,a)$, $(-a,-a)$ and $\left(-\sqrt{3}a,\sqrt{3}a\right)$ are ​the vertices of ​an ​equilateral triangle.
   Show answer

   $A{B}^2=B{C}^2=C{A}^2=8a^2$, so $AB=BC=CA=2\sqrt{2}|a|$. All three sides are equal, so the points are the vertices of an equilateral triangle. Hence proved.

   Step-by-step solution →
3. Q163 CBSE 2025
   Using ​distance formula, ​prove ​that the ​points $(1,5)$, $(2,3)$ and $(3,1)$ ​are ​collinear.
   Show answer

   $AB=\sqrt{5}$, $BC=\sqrt{5}$ and $AC=2\sqrt{5}$, so $AB+BC=AC$. Hence the points $(1,5)$, $(2,3)$ and $(3,1)$ are collinear.

   Step-by-step solution →
4. Q164 CBSE 2025
   If the ​mid-point ​of the line ​segment joining the points $A(3,4)$ ​and $B(k,6)$ ​is $P(x,y)$ ​and $x+y-10=0$, find ​the ​value of $k$.
   Show answer

   The value of $k$ is $7$.

   Step-by-step solution →
5. Q165 CBSE 2025
   In what ​ratio ​does ​the point $(2,p)$ ​divide the ​line segment joining the ​points $(-4,2)$ ​and $(5,-1)$? Also, find the value of '$p$'.
   Show answer

   The point $(2,p)$ divides the segment in the ratio $2:1$ (internally), and $p=0$.

   Step-by-step solution →
6. Q166 CBSE 2025
   The ​points $A(3,6)$, $B(k,2)$ and $C(6,2)$ are the ​vertices ​of a ​right ​triangle $ABC$ right-angled at $B$. Find the ​value of $k$.
   Show answer

   $k=3$

   Step-by-step solution →
7. Q167 CBSE 2025
   Find ​the ​coordinates of the points which divide the ​line segment joining $A(-2,2)$ ​and $B(2,8)$ ​into ​four equal parts.
   Show answer

   The points dividing $AB$ into four equal parts are $P\left(-1,\frac{7}{2}\right)$, $Q(0,5)$ and $R\left(1,\frac{13}{2}\right)$.

   Step-by-step solution →
8. Q168 CBSE 2025
   If the ​point $Q(0,1)$ is ​equidistant ​from points $P(5,-3)$ and $R(x,6)$, then find ​the ​value/s ​of $x$. Also, find ​the ​distance $QR$ and $PR$.
   Show answer

   $x=4$ or $x=-4$; $QR=\sqrt{41}$ units (for both values of $x$); $PR=\sqrt{82}$ units when $x=4$, and $PR=9\sqrt{2}$ units when $x=-4$.

   Step-by-step solution →
9. Q169 CBSE 2025
   $P(x,y),Q(-2,-3)$ and $R(2,3)$ ​are ​the vertices ​of ​a right triangle $PQR$ ​right angled at $P$. Find the relationship between $x$ and $y$. Hence, find all ​possible values of $x$ for which $y=2$.
   Show answer

   The relationship is $x^2+y^2=13$; when $y=2$, $x=3$ or $x=-3$.

   Step-by-step solution →
10. Q170 CBSE 2025
    Find ​the ​ratio ​in which ​the ​y-axis divides the line segment joining the points $(5,-6)$ and $(-1,-4)$. Also find the point ​of ​intersection.
    Show answer

    The $y$-axis divides the segment joining $(5,-6)$ and $(-1,-4)$ in the ratio $5:1$, and the point of intersection is $\left(0,-\frac{13}{3}\right)$.

    Step-by-step solution →
11. Q171 CBSE 2025
    If the mid-point ​of the line ​segment ​joining ​the points $A(3,4)$ and $B(k,6)$ ​is $P(x,y)$ and $x+y-10=0$, then find ​the value of $k$.
    Show answer

    The value of $k$ is $7$.

    Step-by-step solution →
12. Q172 CBSE 2025
    If the points $A(6,1)$, $B(p,2)$, $C(9,4)$ and $D(7,q)$ ​are ​the vertices of a parallelogram $ABCD$, ​then find ​the ​values of $p$ ​and $q$. Hence, check ​whether $ABCD$ is a rectangle or not.
    Show answer

    $p=8$ and $q=3$. The diagonals $AC=3\sqrt{2}$ and $BD=\sqrt{2}$ are not equal, so $ABCD$ is not a rectangle.

    Step-by-step solution →
13. Q173 CBSE 2025
    Find ​a ​relation between $x$ and $y$ such that $P(x,y)$ is equidistant ​from the points $A(3,5)$ and $B(7,1)$. ​Hence, ​write ​the coordinates of the points on $x$-axis and $y$-axis which are equidistant from points $A$ and $B$.
    Show answer

    The required relation is $x-y=2$, that is, $x-y-2=0$; the point on the $x$-axis equidistant from $A$ and $B$ is $(2,0)$, and the point on the $y$-axis is $(0,-2)$.

    Step-by-step solution →
14. Q174 CBSE 2025
    If points $A(-5,y)$, $B(2,-2)$, $C(8,4)$ and $D(x,5)$ ​taken ​in order, form a parallelogram $ABCD$, then ​find the ​values of $x$ ​and $y$. ​Hence, find lengths ​of sides ​of the parallelogram.

    Show answer

    $x=1$ and $y=-1$; the sides are $AB=CD=5\sqrt{2}$ units and $BC=DA=6\sqrt{2}$ units.

    Step-by-step solution →
15. Q175 CBSE 2025
    $A(6,-3)$, $B(0,5)$ and $C(-2,1)$ are vertices ​of $△ABC$. ​Points $P(3,1)$ ​and $Q(2,-1)$ ​lie ​on sides $AB$ ​and $AC$ ​respectively. Check whether $\frac{AP}{PB}=\frac{AQ}{QC}$.
    Show answer

    Yes. $AP=PB=5$ and $AQ=QC=2\sqrt{5}$, so $\frac{AP}{PB}=1$ and $\frac{AQ}{QC}=1$. Hence $\frac{AP}{PB}=\frac{AQ}{QC}$.

    Step-by-step solution →
16. Q176 CBSE 2025
    Prove ​that $P(3,-3)$, $Q(5,-2)$, $R(6,0)$ and $S(4,-1)$ are ​the vertices ​of ​a ​rhombus $PQRS$. Also, find if it is ​a square or ​not.
    Show answer

    All four sides equal $\sqrt{5}$, so $PQRS$ is a rhombus. Its diagonals $PR=3\sqrt{2}$ and $QS=\sqrt{2}$ are unequal, so it is not a square.

    Step-by-step solution →
17. Q177 CBSE 2025
    Find the ​ratio ​in ​which ​the point $(-5,p)$ divides the ​line ​segment joining $(-7,-2)$ ​and $(-4,5)$. Hence, ​find the value of $p$.
    Show answer

    The point divides the segment internally in the ratio $2:1$, and $p=\frac{8}{3}$.

    Step-by-step solution →
18. Q178 CBSE 2025
    A park ​is ​designed in the form of a circle with centre $O$. ​There is a ​fountain at the centre of the park and ​the entry gate ​is at point $A$ on the boundary of the park. Let the coordinates of centre $O$ ​be $(2a-1,7)$ and the coordinates of point $A$ be $(-3,-1)$. Based on the above information, answer the following questions:

    1. (i)If $a=1$, determine the radius of the circular park.[1]
    2. (ii)If the centre of park is on $y$-axis, then find the value of $a$.[1]
    3. (iii)(a)If the radius of the circular park is $10$ units, then find the possible values of $a$.[2]
    4. (iii)(b)If $B$ is another point on the boundary of the park such that $OB\perp OA$, and the radius of the circular park is $10$ units, then determine the length of the chord $AB$.[2]
    Show answer

    (i) The radius is $4\sqrt{5}$ units; (ii) $a=\frac{1}{2}$; (iii)(a) $a=2$ or $a=-4$; OR (iii)(b) $AB=10\sqrt{2}$ units.

    Step-by-step solution →
19. Q179 CBSE 2025
    Case Study - 3: There is ​a semicircular park in Aman's society. He wishes to plant saplings along the boundary of the park. There is a borewell at the centre $O$ ​of the park along the diameter $AB$ as ​shown ​in the figure below. Based on ​the ​above information, answer the ​following questions:

    1. iFind the coordinates of point $O$.[1]
    2. iiFind the radius of the semicircular park.[1]
    3. iii(a)One sapling is kept at point $C(12,y)$. Find the coordinates of $C$.[2]
    4. iii(b)One sapling is kept at point $P$ along $AB$ so that $PA=\frac{1}{3}PB$. Find the coordinates of $P$.[2]
    Show answer

    (i) $O=(12,3)$; (ii) Radius $=10$ units; (iii)(a) $C=(12,-7)$; (iii)(b) $P=(7,3)$.

    Step-by-step solution →
20. Q180 CBSE 2025
    A field is in the form of a rectangle. The coordinates of the rectangular field ABCD are $A(10,10)$, $B(40,10)$, $C(40,50)$ and $D(x,y)$. Anil and Anita, two friends decided to have a race. ​Anita started from point $A$ and moved to point $E$ along the diagonal $AC$, where $E$ is the ​point ​of intersection of both the diagonals of $ABCD$. ​From point $E$, she moved to point $B$ along the other diagonal $DB$ and then moved back to ​point $A$ along $BA$. While Anil started from point $C$ ​and ran to point $A$ via $D$ along the boundary of ​the ​field.

    1. (i)Find the coordinates of point $E$.[1]
    2. (ii)Find the distance between the points $B$ and $C$.[1]
    3. (iii)(a)Find the coordinates of point $D$ and the distance $BD$.[2]
    4. (iii)(b)Find the total distance travelled by Anita.[2]
    Show answer

    (i) $E(25,30)$; (ii) $BC=40$ units; (iii)(a) $D(10,50)$ and $BD=50$ units; (iii)(b) Anita travelled $AE+EB+BA=25+25+30=80$ units.

    Step-by-step solution →