Co-ordinate Geometry questions in CAT essentially tests the concepts of geometry and algebra. We have compiled a List of Top 15 Co-ordinate Geometry Questions for Practice with Video Solutions. Each question has a detailed video and text explanation. You could check the CAT Previous Papers for more practice and knowing the type of questions being asked in the exam. About 1-2 questions are asked each year and taking CAT mock tests regularly will familiarize you with the exam pattern and boost your confidence. Keep practicing and stay consistent!
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1. Coordinate Geometry - Straight Lines
Coordinate geometry formulae:
The distance between two points with coordinates $$(x_1, y_1), (x_2, y_2)$$ is given by $$ d = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$$
Mid point between two points $$A\left(x_1,y_1\right)$$ and $$B\left(x_2,y_2\right)$$ is $$\left(\frac{\left(x_1+x_2\right)}{2},\frac{\left(y_1+y_2\right)}{2}\right)$$
Coordinates of a point P that divides the line joining $$A\left(x_1,y_1\right)$$ and $$B\left(x_2,y_2\right)$$ internally in the ratio l:m : $$\left(\frac{\left(lx_2+mx_1\right)}{l+m},\frac{\left(ly_2+my_1\right)}{l+m}\right)$$.
Coordinates of a point P that divides the line joining $$A\left(x_1,y_1\right)$$ and $$B\left(x_2,y_2\right)$$ externally in the ratio l:m : $$\left(\frac{\left(lx_2-mx_1\right)}{l-m},\frac{\left(ly_2-my_1\right)}{l-m}\right)$$.
A line can be defined as $$y=mx+c$$ where m is the slope of the line and c is the y-intercept.
Slope $$m=\frac{\left(y_2-y_1\right)}{x_2-x_1}$$. Here, if $$x_2=x_1$$, then the two lines are perpendicular to each other.
When two lines are parallel, their slopes are equal i.e $$m_1=m_2$$
When two lines are perpendicular, product of their slopes = -1 i.e $$m_1*m_2=-1$$
If a and b are the x and y intercept of a line then $$\frac{x}{a}+\frac{y}{b}=1$$
If two intersecting lines have slopes $$m_1$$ and $$m_2$$, then the angle between the two lines will be $$\tan\theta\ =\frac{\left(m_1-m_2\right)}{1+m_1m_2}$$.
The length of perpendicular from a point $$\left(X_1,Y_1\right)$$ on the line AX+BY+C=0 is $$\frac{\left(AX_1+BY_1+C\right)}{\sqrt{\ A^2+B^2}}$$.
The distance between two parallel lines Ax+By+C1 = 0 and Ax+By+C2= 0 is $$\left|\frac{C_1-C_2}{\sqrt{\ A^2+B^2}}\right|$$
Image of the point (m,n) in the line ax + by + c = 0 is given by $$\dfrac{\left(x-m\right)}{a}=\dfrac{\left(y-n\right)}{b}=-\dfrac{2\left(am+bn+c\right)}{a^2+b^2}$$
Let C be the circle $$x^{2} + y^{2} + 4x - 6y - 3 = 0$$ 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 $$60^{\circ}$$. Then, the point at which L touches the line $$x$$ = 6 is
correct answer:-2
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
correct answer:-4
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
correct answer:-4
The vertices of a triangle are (0,0), (4,0) and (3,9). The area of the circle passing through these three points is
correct answer:-4
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
correct answer:-9
Given an equilateral triangle T1 with side 24 cm, a second triangle T2 is formed by joining the midpoints of the sides of T1. Then a third triangle T3 is formed by joining the midpoints of the sides of T2. If this process of forming triangles is continued, the sum of the areas, in sq cm, of infinitely many such triangles T1, T2, T3,... will be
correct answer:-4
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
correct answer:-2
The area of the closed region bounded by the equation I x I + I y I = 2 in the two-dimensional plane is
correct answer:-3
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
correct answer:-4
The shortest distance of the point $$(\frac{1}{2},1)$$ from the curve y = I x -1I + I x + 1I is
correct answer:-1
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
correct answer:-1
The area of the triangle whose vertices are (a,a), (a + 1, a + 1) and (a + 2, a) is
[CAT 2002]
correct answer:-2
ABCD is a rhombus with the diagonals AC and BD intersection at the origin on the x-y plane. The equation of the straight line AD is x + y = 1. What is the equation of BC?
correct answer:-1
The points of intersection of three lines $$2x+3y-5=0, 5x-7y+2=0$$ and $$9x-5y-4=0$$
correct answer:-4
What is the distance between the points A(3, 8) and B(-2,-7)?
correct answer:-3