Previous Year CAT Questions for Geometry

1. CAT 2024 QA Slot 1 | Geometry - Coordinate Geometry CAT Question

In the XY-plane, the area, in sq. units, of the region defined by the inequalities

y ≥ x + 4 and -4 ≤ x² + y² + 4(x - y) ≤ 0 is

(a)
2π
(b)
π
(c)
4π
(d)
3π

2. CAT 2024 QA Slot 2 | Geometry - Coordinate Geometry CAT Question

The coordinates of the three vertices of a triangle are: (1, 2), (7, 2), and (1, 10). Then the radius of the incircle of the triangle is

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3. CAT 2023 QA Slot 1 | Geometry - Coordinate Geometry CAT Question

Let C be the circle x² + y² + 4x - 6y - 3 = 0 and L be the locus of the point of intersection of a pair of tangents to C with the angle between the two tangents equal to 60 degree. Then, the point at which L touches the line x = 6 is?

(a)
(6, 6)
(b)
(6, 8)
(c)
(6, 3)
(d)
(6, 4)

Answer: Option C

Video Explanation

Text Explanation :

Let C be the circle x² + y² + 4x - 6y - 3 = 0 and L be the locus of the point of intersection of a pair of tangents to C with the angle between the two tangents equal to 60 degree. Then, the point at which L touches the line x = 6 is?

Given, x² + y² + 4x - 6y - 3 = 0
⇒ x² + 4x + 4 + y² - 6y + 9 - 3 - 4 - 9 = 0 [Adding and Subtracting 4 and 9]
⇒ (x + 2)² + (y - 3)² = 16
⇒ (x + 2)² + (y - 3)² = 4²

This represents the equation of a circle with radius = 4 units and center (O) at (-2, 3).

Point L lies on line x = 6, hence x-coordinate of L is 6. Let y-coordinate of L be 'y'.

Let P be the point where tangent from L touches the given circle.

∆LOP is a 30-60-90° triangle.
⇒ OL = 2 × OP = 8

∴ OL = 8 = $\sqrt{(6 - {(- 2))}^{2} + {(3 - y)}^{2}}$
⇒ 64 = (8)² + (3 - y)²
⇒ y = 3

∴ Coordinate of L = (6, 3)

Hence, option (c).

Note: A locus of points is a set of points that all satisfy some given condition or property.
Here L is the locus of the point of intersection of a pair of tangents to C with the angle between the two tangents equal to 60 degree. Hence, L will be a circle bigger than the original circle as shown. Tangents drawn from any point on this circle to the smaller circle will make an angle of 60 degree.

Concept:

Equation of a Circle

4. CAT 2023 QA Slot 2 | Geometry - Coordinate Geometry CAT Question

The area of the quadrilateral bounded by the Y-axis, the line x = 5 and the lines |x - y| - |5 - x| = 2, is

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5. CAT 2022 QA Slot 1 | Geometry - Coordinate Geometry CAT Question

Let ABCD be a parallelogram such that the coordinates of its three vertices A, B, C are (1, 1), (3, 4) and (−2, 8), respectively. Then, the coordinates of the vertex D are

(a)
(4, 5)
(b)
(0, 11)
(c)
(-4, 5)
(d)
(-3, 4)

6. CAT 2020 QA Slot 3 | Geometry - Coordinate Geometry CAT Question

The points (2, 1) and (-3, -4) are opposite vertices of a parallelogram. If the other two vertices lie on the line x + 9y + c = 0, then c is

(a)
15
(b)
12
(c)
13
(d)
14

Answer: Option D

Video Explanation

Text Explanation :

One of the diagonals of the parallelogram connects the points (2, 1) and (-3, -4).

Line x + 9y + c = 0, represents the other diagonal of the parallelogram.

We know, diagonals of a parallelogram bisect each other.

∴ x + 9y + c = 0 should pass through the midpoint of the line joining (2, 1) and (-3, -4).

The midpoint of the line joining (2, 1) and (-3, -4) = $(\frac{2 - 3}{2}, \frac{1 - 4}{2})$ = (-0.5, -1.5)

⇒ Point (-0.5 , -1.5) lies on x + 9y + c = 0.

⇒ -0.5 + 9 × (-1.5) + c = 0

⇒ -0.5 - 13.5 + c = 0

⇒ c = 14.

Hence, option (d).

7. CAT 2020 QA Slot 3 | Geometry - Coordinate Geometry CAT Question

The vertices of a triangle are (0, 0), (4, 0) and (3, 9). The area of the circle passing through these three points is

(a)
123π/7
(b)
205π/9
(c)
14π/3
(d)
12π/5

Answer: Option B

Video Explanation

Text Explanation :

The diagram can be drawn as follows:

Height of the given triangle = 9 and base = 4

∴ Area of the triangle = $\frac{1}{2} \times 9 \times 4$ = 18

We know area of a triangle = abc/4R

(where a, b, and c are the sides of the triangle and R is the circumradius)

a = AB = 4 units
b = BC = $\sqrt{9^{2} + 1^{2}} = \sqrt{82}$ units
c = AC = $\sqrt{3^{2} + 9^{2}} = \sqrt{90}$ units

The sides of the triangle are 4, √82 and √90

⇒ 18 = $\frac{4 \times \sqrt{82} \times \sqrt{90}}{4 R}$

⇒ R = $\frac{\sqrt{205}}{3}$

∴ Area of the circumcircle = πR² = $\frac{205 π}{9}$

Hence, option (b).

8. CAT 2020 QA Slot 3 | Geometry - Coordinate Geometry CAT Question

The area, in sq. units, enclosed by the lines x = 2, y = |x – 2| + 4, the X-axis and the Y-axis is equal to

(a)
6
(b)
10
(c)
8
(d)
12

9. CAT 2019 QA Slot 1 | Geometry - Coordinate Geometry CAT Question

Let S be the set of all points (x, y) in the x-y plane such that |x| + |y| ≤ 2 and |x| ≥ 1. Then, the area, in square units, of the region represented by S equals

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10. CAT 2019 QA Slot 1 | Geometry - Coordinate Geometry CAT Question

Let T be the triangle formed by the straight line 3x + 5y − 45 = 0 and the coordinate axes. Let the circumcircle of T have radius of length L, measured in the same unit as the coordinate axes. Then, the integer closest to L is

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11. CAT 2018 QA Slot 2 | Geometry - Coordinate Geometry CAT Question

A triangle ABC has area 32 sq units and its side BC, of length 8 units, lies on the line x = 4. Then the shortest possible distance between A and the point (0, 0) is

(a)
8 units
(b)
4 units
(c)
2√2 units
(d)
2√4 units

Answer: Option B

Video Explanation

Text Explanation :

The area of the triangle is 32 sq units.

Therefore, 32 = ½ × BC × height

Given, BC = 8

⇒ Height = $\frac{2 \times 32}{8}$ = 8 cm

Therefore Height of the triangle = 8 cm.

Side BC lies on the line x = 4 or it is parallel to y-axis. Therefore the height of the triangle will be parallel to x axis. In order to minimize the distance of point A from the origin, point A should lie on the x-axis as shown.

Hence, point A is at a distance of 8 units from x = 4 and lies on x-axis.

Therefore point A is (-4,0) and the distance of point A from the origin = 4 units.

Hence, option (b).

12. CAT 2017 QA Slot 1 | Geometry - Coordinate Geometry CAT Question

The area of the closed region bounded by the equation | x | + | y | = 2 in the two-dimensional plane is

(a)
4π
(b)
4
(c)
8
(d)
2π

Answer: Option C

Video Explanation

Text Explanation :

The closed region bounded by |ax| + |by| = c in the two-dimensional plane has x-intercepts of ± $\frac{c}{| a |}$ and y-intercepts of ± $\frac{c}{| b |}$ . This is in general a rhombus. In the given question, we have a square which has each of its diagonals as 4.

Area = 1/2 × 4 × 4 = 8.

Hence, option (c).

13. CAT 2017 QA Slot 2 | Geometry - Coordinate Geometry CAT Question

The points (2, 5) and (6, 3) are two end points of a diagonal of a rectangle. If the other diagonal has the equation y = 3x + c, then c is

(a)
-5
(b)
-6
(c)
-7
(d)
-8

Answer: Option D

Video Explanation

Text Explanation :

Let ABCD be the rectangle where A = (2, 5) and C= (6, 3).

We know that AC and BD will be the diagonals of the rectangle. It can be represented using the diagram shown below

In the above figure, O is the point of intersection of the 2 diagonals and the mid point of both the diagonals.

Mid Point of AC = $(\frac{2 + 6}{2}, \frac{5 + 3}{2}) \equiv$ (4, 4)

Now O also lies on diagonal BD [Diagonal of a rectangle bisect each other.]

Hence co-ordinates of O will satisfy the equation

y = 3x + c
∴ 4 = 3 × 4 + c
⇒ c = – 8

Hence, option (d).

14. CAT 2005 QA | Geometry - Coordinate Geometry CAT Question

Consider a triangle drawn on the X-Y plane with its three vertices at (41, 0), (0, 41) and (0, 0), each vertex being represented by its (X, Y) coordinates. The number of points with integer coordinates inside the triangle (excluding all the points on the boundary) is

(a)
780
(b)
800
(c)
820
(d)
741

15. CAT 2002 QA | Geometry - Coordinate Geometry CAT Question

The area of the triangle with the vertices (a, a), (a + 1, a) and (a, a + 2) is

(a)
a³
(b)
1
(c)
0
(d)
None of these

Answer: Option B

Text Explanation :

Area (∆) = $|\begin{matrix} x_{1} & y_{1} & 1 \\ x_{2} & y_{2} & 2 \\ x_{3} & y_{3} & 3 \end{matrix}|$

where x₁ = y₁ = a, x₂ = a + 1, y₂ = a, x₃ = a and y₃ = a + 2

∴ Area (∆) = $\frac{1}{2} |\begin{matrix} a & a & 1 \\ a + 1 & a & 1 \\ a & a + 12 & 1 \end{matrix}|$

∴ Area (∆) = $\frac{1}{2}$ {a[a(1) - (a + 2)(1)] - a[(a + 1)(1) - a(1)] + 1[(a + 1)(a + 2) - a²]} = $\frac{1}{2}$ {- 2a - a + 3a + 2} = 1

Hence, option (b).

16. CAT 2000 QA | Geometry - Coordinate Geometry CAT Question

The area bounded by the three curves |x + y| = 1, |x| = 1, and |y| = 1, is equal to

(a)
4
(b)
3
(c)
2
(d)
1

17. CAT 2000 QA | Geometry - Coordinate Geometry CAT Question

Choose 1; if the question can be answered by using one of the statements alone, but cannot be answered using the other statement alone.
Choose 2; if the question can be answered by using either statement alone.
Choose 3; if the question can be answered by using both statements together, but cannot be answered using either statement alone.
Choose 4; if the question cannot be answered even by using both statements together.

There are two straight lines in the x-y plane with equations ax + by = c , dx + ey = f. Do the two straight lines intersect?

a, b, c, d, e and f are distinct real numbers.
c and f are non-zero.

(a)
1
(b)
2
(c)
3
(d)
4

18. CAT 1999 QA | Geometry - Coordinate Geometry CAT Question

The robot reaches point (6, 6) when a sequence of three instructions is executed, the first of which is a GOTO(x,y) instruction, the second is WALKX(2) and the third is WALKY(4). What are the value of x and y?

(a)
2, 4
(b)
0, 0
(c)
4, 2
(d)
2, 2

19. CAT 1999 QA | Geometry - Coordinate Geometry CAT Question

The robot is initially at (x, y), x > 0 and y < 0. The minimum number of instructions needed to be executed to bring it to the origin (0, 0) if you are prohibited from using the GOTO instruction is

(a)
2
(b)
1
(c)
x + y
(d)
0

20. CAT 1991 QA | Geometry - Coordinate Geometry CAT Question

What is the distance between the points A(3, 8) and B(–2, –7)?

(a)
5√2
(b)
5
(c)
5√10
(d)
10√2

Answer: Option A

Video Explanation

Text Explanation :

The distance between two points (x₁, y₁) and (x₂, y₂) is given as $\sqrt{{(x_{2} - x_{1})}^{2} + {(y_{2} - y_{1})}^{2}}$

Hence, required distance = $\sqrt{{(- 2 - 3)}^{2} + {(- 7 - 8)}^{2}}$ = 5√10.

Hence, option (c).

Geometry - Coordinate Geometry - Previous Year CAT/MBA Questions

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