Coordinate Geometry

CBSE Class 10 — Maths — Previous Year Questions

Q1.MCQ1 mark(2020)Medium

The distance between the points $(a\cos \theta +b\sin \theta ,0)$ and $(0,a\sin \theta -b\cos \theta )$, is:

Q2.MCQ1 mark(2020)Easy

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:

Q3.MCQ1 mark(2020)Easy

The value of $p$, for which the points $A(3,1)$, $B(5,p)$ and $C(7,-5)$ are collinear, is:

Q4.MCQ1 mark(2020)Easy

The point $P$ on x-axis equidistant from the points $A(-1,0)$ and $B(5,0)$ is

Q5.MCQ1 mark(2020)Easy

The co-ordinates of the point which is reflection of point $(-3,5)$ in x-axis are

Q6.MCQ1 mark(2020)Easy

If the point $P(6,2)$ divides the line segment joining $A(6,5)$ and $B(4,y)$ in the ratio $3:1$, then the value of $y$ is

Q7.MCQ1 mark(2020)Easy

If $(a,b)$ is the mid-point of the line segment joining the points $A(10,-6)$ and $B(k,4)$ and $a-2b=18$, the value of $k$ is

Q8.MCQ1 mark(2020)Easy

The value of $k$ for which the points $A(0,1)$, $B(2,k)$ and $C(4,-5)$ are collinear is

Q9.VSA1 mark(2020)Easy

$\sqrt{2}$ times the distance between $(0,5)$ and $(-5,0)$ is ________.

Q10.MCQ1 mark(2020)Easy

It is being given that the points $A(1,2)$, $B(0,0)$ and $C(a,b)$ are collinear. Which of the following relations between $a$ and $b$ is true?

Q11.MCQ1 mark(2020)Easy

Point $P\left(\frac{a}{8},4\right)$ is the mid-point of the line segment joining the points $A(-5,2)$ and $B(4,6)$. The value of '$a$' is

Q12.VSA1 mark(2020)Easy

The distance between the points $(a,b)$ and $(-a,-b)$ is ________.

Q13.MCQ1 mark(2020)Easy

The point on the x-axis which is equidistant from $(-4,0)$ and $(10,0)$ is

Q14.MCQ1 mark(2020)Easy

The centre of a circle whose end points of a diameter are $(-6,3)$ and $(6,4)$ is

Q15.MCQ1 mark(2020)Easy

The distance between the points $(m,-n)$ and $(-m,n)$ is

Q16.MCQ1 mark(2020)Easy

$AOBC$ is a rectangle whose three vertices are $A(0,-3)$, $O(0,0)$ and $B(4,0)$. The length of its diagonal is ________.

Q17.MCQ1 mark(2023)Medium

In what ratio does the x-axis divide the line segment joining the points $A(3,6)$ and $B(-12,-3)$?
(A) $1:2$
(B) $1:4$
(C) $4:1$
(D) $2:1$

Q18.MCQ1 mark(2023)Easy

The distance between the points $(0,2\sqrt{5})$ and $(-2\sqrt{5},0)$ is
(A) $2\sqrt{10}$ units
(B) $4\sqrt{10}$ units
(C) $2\sqrt{20}$ units
(D) $0$

Q19.MCQ1 mark(2023)Easy

The coordinates of the vertex A of a rectangle ABCD whose three vertices are given as $B(0,0)$, $C(3,0)$ and $D(0,4)$ are:
(a) $(4,0)$
(b) $(0,3)$
(c) $(3,4)$
(d) $(4,3)$

Q20.MCQ1 mark(2023)Easy

The area of the triangle formed by the line $\frac{x}{a}+\frac{y}{b}=1$ with the coordinate axes is:
(a) $ab$
(b) $\frac{1}{2}ab$
(c) $\frac{1}{4}ab$
(d) $2ab$

Q21.MCQ1 mark(2023)Easy

The distance of the point $(-1,7)$ from x-axis is:

Q22.MCQ1 mark(2023)Easy

The distance of the point $(-6,8)$ from origin is:

Q23.Case-Study1 mark(2023)Easy

Jagdish has a field which is in the shape of a right angled triangle AQC. He wants to leave a space in the form of a square PQRS inside the field for growing wheat and the remaining for growing vegetables (as shown in the figure). In the field, there is a pole marked as O.



Based on the above information, answer the following questions:

(i) Taking O as origin, coordinates of P are $(-200,0)$ and of Q are $(200,0)$. PQRS being a square, what are the coordinates of R and S?

Q24.Case-Study1 mark(2023)Medium

(iii) If S divides CA in the ratio $K:1$, what is the value of K, where point A is $(200,800)$?

Q25.MCQ1 mark(2023)Easy

The distance between the points $P\left(-\frac{11}{3},5\right)$ and $Q\left(-\frac{2}{3},5\right)$ is:

Q26.MCQ1 mark(2023)Easy

The distance of the point $(-6,8)$ from $x$-axis is

Q27.MCQ1 mark(2024)Easy

$AD$ is a median of $△ABC$ with vertices $A(5,-6)$, $B(6,4)$ and $C(0,0)$. Length $AD$ is equal to:

Q28.MCQ1 mark(2024)Easy

If the distance between the points $(3,-5)$ and $(x,-5)$ is $15$ units, then the values of $x$ are:

Q29.MCQ1 mark(2024)Easy

The centre of a circle is at $(2,0)$. If one end of a diameter is at $(6,0)$, then the other end is at:

Q30.MCQ1 mark(2024)Easy

Point $P$ divides the line segment joining the points $A(4,-5)$ and $B(1,2)$ in the ratio $5:2$. Co-ordinates of point $P$ are

Q31.MCQ1 mark(2024)Easy

$XOYZ$ is a rectangle with vertices $X(-3,0)$, $O(0,0)$, $Y(0,4)$ and $Z(x,y)$. The length of its each diagonal is

Q32.MCQ1 mark(2024)Easy

The fourth vertex $D$ of a parallelogram $ABCD$ whose three vertices are $A(-2,3)$, $B(6,7)$ and $C(8,3)$ is:

Q33.Assertion-Reason1 mark(2024)Medium

**Assertion (A):** Mid-point of a line segment divides the line segment in the ratio $1:1$.

**Reason (R):** The ratio in which the point $(-3,k)$ divides the line segment joining the points $(-5,4)$ and $(-2,3)$ is $1:2$.

Q34.MCQ1 mark(2025)Easy

The mid-point of the line segment joining the points $P(-4,5)$ and $Q(4,6)$ lies on:

Q35.MCQ1 mark(2025)Easy

The coordinates of the end points of a diameter of a circle are $(5,-2)$ and $(5,2)$. The length of the radius of the circle is:

Q36.MCQ1 mark(2025)Easy

The points $(-5,0)$, $(5,0)$ and $(0,4)$ are the vertices of a triangle which is also:

Q37.MCQ1 mark(2025)Easy

The equation of a line parallel to the x-axis and at a distance of 3 units below x-axis is:

Q38.MCQ1 mark(2025)Easy

The line represented by $\frac{x}{4}+\frac{y}{6}=1$, intersects x-axis and y-axis respectively at P and Q. The coordinates of the mid-point of line segment PQ are:

Q39.MCQ1 mark(2025)Easy

Two of the vertices of $△PQR$ are $P(-1,5)$ and $Q(5,2)$. The coordinates of a point which divides PQ in the ratio $2:1$ are:

Q40.MCQ1 mark(2025)Easy

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:

Q41.MCQ1 mark(2025)Easy

The distance of a point A from x-axis is 3 units. Which of the following cannot be coordinates of the point A?

Q42.VSA2 marks(2023)Easy

Show that the points $(-2,3)$, $(8,3)$ and $(6,7)$ are the vertices of a right-angled triangle.

Q43.Case-Study2 marks(2023)Easy

(ii) (a) What is the area of square PQRS?

Q44.Case-Study2 marks(2023)Easy

(ii) (b) What is the length of diagonal PR in square PQRS?

Q45.VSA2 marks(2023)Easy

The line segment joining the points $A(4,-5)$ and $B(4,5)$ is divided by the point $P$ such that $AP:AB=2:5$. Find the coordinates of $P$.

Q46.VSA2 marks(2023)Easy

Point $P(x,y)$ is equidistant from points $A(5,1)$ and $B(1,5)$. Prove that $x=y$.

Q47.VSA2 marks(2023)Easy

Find the ratio in which $y$-axis divides the line segment joining the points $(5,-6)$ and $(-1,-4)$.

Q48.VSA2 marks(2023)Easy

A line intersects $y$-axis and $x$-axis at point $P$ and $Q$, respectively. If $R(2,5)$ is the mid-point of line segment $PQ$, then find the coordinates of $P$ and $Q$.

Q49.VSA2 marks(2024)Easy

Find the type of triangle $ABC$ formed whose vertices are $A(1,0)$, $B(-5,0)$ and $C(-2,5)$.

Q50.VSA2 marks(2024)Medium

In what ratio is the line segment joining the points $(3,-5)$ and $(-1,6)$ divided by the line $y=x$?

Q51.VSA2 marks(2024)Medium

$A(3,0)$, $B(6,4)$ and $C(-1,3)$ are vertices of a triangle $ABC$. Find length of its median $BE$.

Q52.VSA2 marks(2024)Easy

Find the ratio in which the point $P(-4,6)$ divides the line segment joining the points $A(-6,10)$ and $B(3,-8)$.

Q53.VSA2 marks(2024)Easy

Prove that the points $(3,0)$, $(6,4)$ and $(-1,3)$ are the vertices of an isosceles triangle.

Q54.VSA2 marks(2025)Medium

The coordinates of the centre of a circle are $(2a,a-7)$. Find the value(s) of $a$ if the circle passes through the point $(11,-9)$ and has diameter $10\sqrt{2}$ units.

Q55.VSA2 marks(2025)Medium

Find the coordinates of the point C which lies on the line AB produced such that $AC=2BC$, where coordinates of points A and B are $(-1,7)$ and $(4,-3)$ respectively.

Q56.SA3 marks(2020)Easy

Find the area of triangle $PQR$ formed by the points $P(-5,7)$, $Q(-4,-5)$ and $R(4,5)$.

Q57.SA3 marks(2020)Medium

If the point $C(-1,2)$ divides internally the line segment joining $A(2,5)$ and $B(x,y)$ in the ratio $3:4$, find the coordinates of $B$.

Q58.SA3 marks(2020)Easy

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 $k$.

Q59.SA3 marks(2020)Medium

Find the area of triangle $ABC$ with $A(1,-4)$ and the mid-points of sides through $A$ being $(2,-1)$ and $(0,-1)$.

Q60.SA3 marks(2020)Medium

In what ratio does the point $P(-4,y)$ divide the line segment joining the points $A(-6,10)$ and $B(3,-8)$ if it lies on $AB$. Hence find the value of $y$.

Q61.SA3 marks(2020)Medium

Find the co-ordinates of the points of trisection of the line segment joining the points $(3,-1)$ and $(6,8)$.

Q62.SA3 marks(2020)Medium

Find the area of a quadrilateral $ABCD$ having vertices at $A(1,2)$, $B(1,0)$, $C(4,0)$ and $D(4,4)$.

Q63.SA3 marks(2020)Medium

Find the ratio in which the y-axis divides the line segment joining the points $(6,-4)$ and $(-2,-7)$. Also find the point of intersection.

Q64.SA3 marks(2020)Medium

Show that the points $(7,10)$, $(-2,5)$ and $(3,-4)$ are the vertices of an isosceles right triangle.

Q65.SA3 marks(2023)Hard

If $(-5,3)$ and $(5,3)$ are two vertices of an equilateral triangle, then find coordinates of the third vertex, given that origin lies inside the triangle. (Take $\sqrt{3}=1.7$)

Q66.SA3 marks(2023)Easy

If $Q(0,1)$ is equidistant from $P(5,-3)$ and $R(x,6)$, find the values of $x$.

Q67.SA3 marks(2023)Medium

Find the ratio in which the line segment joining the points $A(6,3)$ and $B(-2,-5)$ is divided by $x$-axis.

Q68.SA3 marks(2024)Medium

Find the ratio in which the point $\left(\frac{8}{5},y\right)$ divides the line segment joining the points $(1,2)$ and $(2,3)$. Also, find the value of $y$.

Q69.SA3 marks(2024)Medium

ABCD is a rectangle formed by the points $A(-1,-1)$, $B(-1,6)$, $C(3,6)$ and $D(3,-1)$. P, Q, R and S are mid-points of sides AB, BC, CD and DA respectively. Show that diagonals of the quadrilateral PQRS bisect each other.

Q70.SA3 marks(2025)Medium

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.

Q71.SA3 marks(2025)Easy

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 $k$.

Q72.SA3 marks(2025)Easy

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 $k$.

Q73.SA3 marks(2025)Medium

Find the coordinates of the points which divide the line segment joining $A(-2,2)$ and $B(2,8)$ into four equal parts.

Q74.SA3 marks(2025)Medium

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 $y=2$.

Q75.Case-Study4 marks(2024)Medium

Ryan, from a very young age, was fascinated by the twinkling of stars and the vastness of space. He always dreamt of becoming an astronaut one day. So he started to sketch his own rocket designs on the graph sheet. One such design is given below:



Based on the above, answer the following questions:
(i) Find the mid-point of the segment joining $F$ and $G$.
(ii) (a) What is the distance between the points $A$ and $C$?

**OR**

(ii) (b) Find the coordinates of the point which divides the line segment joining the points $A$ and $B$ in the ratio $1:3$ internally.
(iii) What are the coordinates of the point $D$?

Q76.Case-Study4 marks(2024)Medium

Ryan, from a very young age, was fascinated by the twinkling of stars and the vastness of space. He always dreamt of becoming an astronaut one day. So he started to sketch his own rocket designs on the graph sheet. One such design is given below:



Based on the above, answer the following questions:
(i) Find the mid-point of the segment joining $F$ and $G$.
(ii) (b) Find the coordinates of the point which divides the line segment joining the points $A$ and $B$ in the ratio $1:3$ internally.
(iii) What are the coordinates of the point $D$?

Q77.Case-Study4 marks(2024)Medium

A garden is in the shape of a square. The gardener grew saplings of Ashoka tree on the boundary of the garden at the distance of 1 m from each other. He wants to decorate the garden with rose plants. He chose a triangular region inside the garden to grow rose plants. In the above situation, the gardener took help from the students of class 10. They made a chart for it which looks like the given figure.



Based on the above, answer the following questions:
(i) If $A$ is taken as origin, what are the coordinates of the vertices of $△PQR$?
(ii) (a) Find distances $PQ$ and $QR$.
OR
(ii) (b) Find the coordinates of the point which divides the line segment joining points $P$ and $R$ in the ratio $2:1$ internally.
(iii) Find out if $△PQR$ is an isosceles triangle.

Q78.Case-Study4 marks(2024)Medium

A garden is in the shape of a square. The gardener grew saplings of Ashoka tree on the boundary of the garden at the distance of 1 m from each other. He wants to decorate the garden with rose plants. He chose a triangular region inside the garden to grow rose plants. In the above situation, the gardener took help from the students of class 10. They made a chart for it which looks like the given figure.



Based on the above, answer the following questions:
(i) If $A$ is taken as origin, what are the coordinates of the vertices of $△PQR$?
(ii) (b) Find the coordinates of the point which divides the line segment joining points $P$ and $R$ in the ratio $2:1$ internally.
(iii) Find out if $△PQR$ is an isosceles triangle.