Sample QuestionsCo-ordinate Geometry questions
One sample from each question group in this chapter. Select any group above to see the full set with answer keys.
If the line segment joining the points $(3, -4)$ and $(1, 2)$ is trisected at points $P(a, -2)$ and $\text{Q}\Big(\frac{5}{3},\text{b}\Big).$ Then,
- A
$\text{a}=\frac{8}{3},\text{b}=\frac{2}{3}$
- ✓
$\text{a}=\frac{7}{3},\text{b}=0$
- C
$\text{a}=\frac{1}{3},\text{b}=1$
- D
$\text{a}=\frac{2}{3},\text{b}=\frac{1}{3}$
Answer: B.
View full solution →If points $(t, 2t), (-2, 6)$ and $(3, 1)$ are collinear, then $t =$
- A
$\frac{3}{4}$
- ✓
$\frac{4}{3}$
- C
$\frac{5}{3}$
- D
$\frac{3}{5}$
Answer: B.
View full solution →The distance between the points $(\text{a}\cos25^\circ,0)$ and $(0,\text{a}\cos65^\circ)$ is:
Answer: A.
View full solution →The coordinates of the fourth vertex of the rectangle formed by the points (0, 0), (2, 0), (0, 3) are,
Answer: C.
View full solution →A line segment is of length $10$ units. If the coordinates of its one end are $(2, -3)$ and the abscissa of the other end is $10$, then its ordinate is:
- A
$9, 6$
- ✓
$3, -9$
- C
$-3, 9$
- D
$9, -6$
Answer: B.
View full solution →Statement-1 (A): The distance of the point $(-3,5)$ from the $x$ axis is 3 units.
Statement-2 (R): Abscissa of a point gives the distance of the point from the $y$-axis.
- A
Statement- 1 is true, Statement- 2 is true; Statement- 2 is a correct explanation for Statement-1.
- B
Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.
- C
Statement-1 is true, Statement-2 is false.
- ✓
Statement-1 is false, Statement-2 is true.
Answer: D.
View full solution →Statement-1 (A): The point which divides the line segment joining the points $A(1,2)$ and $B(-1,1)$ internally in the ratio $1: 2$ is $\left(-\frac{1}{3}, \frac{5}{3}\right)$.
Statement-2 (R) : The coordinates of the point which divides the line segment joining the points $A\left(x_1, y_1\right)$ and $B\left(x_2, y_2\right)$ in the ratio $m_1: m_2$ are $\left(\frac{m_1 x_2+m_2 x_1}{m_1+m_2}, \frac{m_1 y_2+m_2 y_1}{m_1+m_2}\right)$.
- A
Statement- 1 is true, Statement- 2 is true; Statement- 2 is a correct explanation for Statement-1.
- B
Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.
- C
Statement-1 is true, Statement-2 is false.
- ✓
Statement-1 is false, Statement-2 is true.
Answer: D.
View full solution →Statement-1 (A): Mid-point of a line segment divides the line segment in the ratio $1: 1$.
Statement-2 (R) : The ratio in which the point $(-3, k)$ divides the line segment joining the points $(-5,4)$ and $(-2,3)$ is $1: 2$.
- A
Statement- 1 is true, Statement- 2 is true; Statement- 2 is a correct explanation for Statement-1.
- B
Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.
- ✓
Statement-1 is true, Statement-2 is false.
- D
Statement-1 is false, Statement-2 is true.
Answer: C.
View full solution →Statement-1 (A): If the points $A(4,3)$ and $B(x, 5)$ lie on a circle with centre $O(2,3)$, then the value of $x$ is 2 .
Statement-2 (R) : Centre of the circle is the mid-point of each chord of the circle.
- A
Statement-1 and Statement-2 are True; Statement- 2 is a correct explanation for Statement-1.
- B
Statement -1 and Statement- 2 are True; Statement- 2 is not a correct explanation for Statement-1,
- ✓
Statement-1 is True, Statement-2 is False.
- D
Statement-1 is False, Statement- 2 is True.
Answer: C.
View full solution →Statement-1 (A): Point $P(0,2)$ is the point of intersection of $y$-axis with the line $3 x+2 y=4$.
Statement-2 (R) : The distance of the point $P(0,2)$ from $x$-axis is 2 units.
- A
Statement- 1 is true, Statement- 2 is true; Statement- 2 is a correct explanation for Statement-1.
- ✓
Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.
- C
Statement-1 is true, Statement-2 is false.
- D
Statement-1 is false, Statement-2 is true.
Answer: B.
View full solution →If the centroid of the triangle formed by the points (a, b), (1, a) and (b, 1) is at the origin, then $\frac{a^3+b^3+1}{a b}$ = ____________ .
View full solution →The coordinates of the point equidistant from the vertices O(0, 0) A(6, 0) and B(0, 8) of $\triangle OAB$ are ____________ .
View full solution →If the distance of the point (4, a) from x-axis is half its distance from y-axis, then ____________ .
The coordinates of the point which is equidistant from the vertices of the triangle formed by the points O(0, 0) A(a, 0) and B(0, b) are ____________ .
It (2, -2) and (5, 2) are opposite vertices of a square, then the length of the side of the square is ____________ .
Points A (-1, y) and B (5, 7) lie on a circle with centre O (2,-3y) such that AB is a diameter of the circle. Find the value of y. Also, find the radius of the circle.
A (3, 0), B (6, 4) and C (-1, 3) are vertices of a triangle ABC. Find the length of its median BE.
In what ratio is the line segment joining the points (3, -5) and (-1, 6) divided by the line y = x?
Find the type of triangle ABC formed whose vertices are A (1, 0), B (-5, 0) and C (-2, 5).
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.
What is the distance between the points A(c, 0) and B(0, -c)?
Find the value of a so that the point $(3, a)$ lies on the line represented by $2 x-3 y+5=0$
View full solution →On which axis do the following points lie?
S(0, 5).
Find the distance between the points $\Big(\frac{-8}{5},2\Big)$ and $\Big(\frac{2}{5},2\Big).$
View full solution →If P(2, 6) is the mid-point of the line segment joining A(6, 5) and B(4, y), find y.
The area of a triangle is 5sq units. Two of its vertices are $(2, 1)$ and $(3, -2)$. If the third vertex is $\Big(\frac{7}{2},\text{y}\Big),$ find the value of y.
View full solution →Find the distance of the point (1, 2) from the mid-point of the line segment joining the points (6, 8) and (2, 4).
In the seating arrangement of desks in a classroom three students Rohini, Sandhya and Bina are seated at A(3, 1), B(6, 4) and C(8, 6). Do you think they are seated in a line?
Show that the points (-3, 2), (-5, -5), (2, -3) and (4, 4) are the vertices of a rhombus. Find the area of this rhombus.
If the points A(-1, -4), B(b, c) and C(5, -1) are collinear and 2b + c = 4, find the values of b and c.
The three vertices of a parallelogram are (3, 4) (3, 8) and (9, 8). Find the fourth vertex.
The points A(2, 9), B(a, 5) and C(5, 5) are the vertices of a triangle ABC right angled at B. Find the values of a and hence the area of $\triangle\text{ABC}.$
View full solution →Find the lengths of the medians of a triangle whose vertices are A(-1, 3), B(1, -1) and C(5, 1).
Find the ratio in which the point (2, y) divides the line segment joining the points A(-2, 2) and B(3, 7). Also, find the value of y.
Prove that the points (-4, -1), (-2, 4), (4, 0) and (2, 3) are the vertices of a rectangle.
Drones are used by military for surveillance purposes. These days, drones are also used by individual entrepreneurs, SMEs and large companies to accomplish various other tasks.
Adrone is flying over a rectangular field with vertices at $A(-100,0), B(100,0), C(100,150)$ and $D(-100,150)$. The drone captures an image at a location $(x, y)$.
(i) Find the dimensions of the rectangular field.
(ii) Find the distance between points A and C .
(iii) (a) If a drone captures the image of an object $P(x, y)$ on the rectangular field, find the relation between $x$ and $y$ such that $P A=P C$.
OR
(b) If a drone captures the image of an object at a point $Q$ whose $x$ coordinate is 0 and it is equidistant from points $A$ and $D$, find the coordinates of $Q$. View full solution →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 Fig.
(i) If $A$ is taken as origin, what are the coordinates of the vertices of $\triangle P Q R$ ?
(ii) (a) Find the distances $P Q$ and $Q R$.
OR
(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 $\triangle P Q R$ is an isoscales triangle. View full solution →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:
(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
(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 moint D? View full solution →Jagdish has a field which is in the shape of a right angled triangle $A Q C$. He wants to leave a space in the form of a square $P Q R S$ inside the field for growing wheat and the remaining for growing vegetables (as shown in the Fig. 6.10). In the field, there is a pole marked as $O$.
(i) Taking $O$ as origin, coordinates of $P$ are $(-200,0)$ and $Q$ are $(200,0) . P Q R S$ being a square, what are the coordinates of $R$ and $S$ ?
(ii) What is the area of square $P Q R S$ ?
(iii) What is the length of diagonal $P R$ in square $P Q R S$ ?
(iv) If $S$ divides $C \Lambda$ in the ratio $k: 1$, what is the value of $k$, where point $\Lambda$ is $(200,800)$ ? View full solution →Tharunya was thrilled to know that the football tournament is fixed with a monthly time frame from 20th July to 20th August 2023 and for the first time in the FIFA Women's World Cup's history, two mations host in 10 venues. Her father fell that the game can be belter understood if the position of players is represented as points on a coordinate plane.
(i) At an inslance, the mid fielders and forward formed a parallelogram. Find the position of the central mid fielder (D) if the position of other players who formed the parallelogram are: $A(1,2), B(4,3)$ and $C(6,6)$.
(ii) Check if the Goal keeper $G(-3,5)$, Sweeper $H(3,1)$ and Wing-back $K(0,3)$ fall on a same straight line.
(iii) Check if the Full-back $J(5,-3)$ and centre-back I $(-4,6)$ are equidistant from forward $C(0,1)$ and if $C$ is the mid-point of IJ.
(iv) If Defensive mid fielder $\Lambda(1,4)$, Altacking mid fielder $B(2,-3)$ and Striker $E(a, b)$ lie on the same straight line and B is equidistant from $\triangle$ and $E$, find the position of $E$. View full solution →