K = 0 is rejected as at k =0 will give the point at A which is also not allowed. Now, we know that if the x,y coordinates of a point are integer, the sum will also be an integerĪs per the question we need to exclude all the values lying on the boundary of triangle, k can take all values from 1 to 40 only. Solution: Equation of the line will be of the form => x + y = 41. The number of points with integer coordinates inside the triangle (excluding all the points on the boundary) is: Imp: The main point to note here is that area will be independent of a.Įxample 4: Consider a triangle drawn on the X – Y plane with its three vertices of (41,0), (0,41) and (0,0), each vertex being represented by its (X,Y) coordinates. Solution: Let a = 0, Thus the three vertices of the triangle becomes (1, 1) (0, 0) and (2, 0) This suggests that the three lines intersect at a common point and hence coincident.Įxample 3: The area of the triangle whose vertices are (a + 1, a + 1), (a, a) and (a+2, a) is If we substitute (1,1) in the third equation Y=(9X/5 – 4/5), we find that it also satisfies the equation. Solving the first two equations we get X=1 and Y = 1. Thus, lines are not parallel to each other.

Thus, lines are not perpendicular to each other.Īlso, slopes of the no two lines is same. We see above that the product of slopes of none of the lines is -1. Therefore, the slopes of the three lines are -2/3, 5/7, 9/5 and their Y intercepts are 5/3, 2/7 and 4/5 respectively. The three lines can be expressed in the y=mx + c format as: Lines are coincident if they at least have one point which satisfies all the equation.Two lines are perpendicular if the product of slopes is -1.Two lines are parallel to each other if their slopes are equal.Solution:To solve the question above, we should remember the properties of the lines for being parallel, perpendicular or intersecting Hence, required distance = sqrt((-2-3) 2 + (-7-8) 2)Įxample 2: The points of intersection of three lines 2X + 3Y – 5=0 and 5X – 7Y + 2=0 and 9X – 5Y – 4 = 0ī.) Are on lines perpendicular to each other Solution: The distance between 2 points (x1, y1) and (x2, y2) is given as Quadrant IV => X is Positive Y is Negative Coordinate Geometry Sample CAT Questions:Įxample 1: What is the distance between the points A (3,8) and B(-2,-7) ? Quadrant III => X is Negative Y is Negative Quadrant II => X is Negative Y is Positive Quadrant I => X is Positive Y is Positive → Intercept form x / a + y / b = 1 (where a and b are x and y intercepts respectively) → Point-slope form y – y1 = m (x – x1 ) (m is the slope of the line) → Slope intercept form y = mx + c (c is y intercept) The length of perpendicular from a point (x1 ,y1 ) on the line AX+BY+C = 0 is.If two intersecting lines have slopes m1 and m2 then the angle between two lines θ will be.The equation of a straight line is y = mx + c, where m is the slope and c is the y-intercept (tan = m, where is the angle that the line makes with the positive X-axis).Where a, b and c are the lengths of the BC, AC and AB respectively.

If I (x, y) is the in-center of triangle ABC, A (x 1, y 1), B (x 2, y 2), C (x 3, y 3), then,.

If G (x, y) is the centroid of triangle ABC, A (x 1, y 1), B (x 2, y 2), C (x 3, y 3), then,.

If point P(x, y) divides the segment AB, where A (x 1, y 1) and B (x 2, y 2), internally in the ratio m: n, then,.

The distance between the points (x 1, y 1) and (x 2, y 2) is.

If m1 and m2 are slopes of two parallel lines, then m1=m2.

The slopes of two parallel lines are always equal.The product of the slopes of two perpendicular lines is –1.We will use the below picture as a reference for the formulas. Now, Let us have a look at some formulas for coordinate geometry. The axes intersect at the origin which is the point (0,0). The x-axis is the horizontal number line and the y-axis is the vertical number line. In coordinate geometry, we can use graphs and coordinates to find measurements and other useful information about geometric figures.Ī coordinate graph is a rectangular grid with two number lines called axes. We can use rulers to measure lines and protractors to measure angles. There are all sorts of ways that we can find the measurements of lines and angles.

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Geometry the same line m1 and m2