1Which of these is the equation of a line passing through two points: \( y = mx + c \), \( \frac{x}{a} + \frac{y}{b} = 1 \), \( y - y_1 = m(x - x_1) \), or \( y - y_1 = \frac{y_2 - y_1}{x_2 - x_1}(x - x_1) \)?
2Which is the equation of the line through \( (2, 3) \) and \( (-1, 0) \): \( x - y + 1 = 0 \), \( x + y + 1 = 0 \), \( x - y - 1 = 0 \), or \( -x - y - 1 = 0 \)?
3Which is the equation of the line through \( (a, 0) \) and \( (0, b) \): \( \frac{x}{b} + \frac{y}{a} = 1 \), \( \frac{x}{a} + \frac{y}{b} = 1 \), \( bx + ay = 0 \), or \( ax + by = 0 \)?
4Which is the equation of the line through \( (-4, -2) \) and \( (-3, 5) \): \( y = 7x + 26 \), \( y = -7x + 26 \), \( y = -7x - 26 \), or \( y = \frac{1}{7}x + 26 \)?
5Describe how the equation of a line can be found when two points on it are given.
6Find the equation of the line shown in graph (क).
7Find the equation of the line shown in graph (ख).
8Find the equation of the line shown in graph (ग).
9Find the equation of the line shown in graph (घ).
10Find the equation of the straight line through \( (2, 3) \) and \( (-1, 0) \).
11Find the equation of the straight line through \( (4, 3) \) and \( (-2, 4) \).
12Find the equation of the straight line through \( (6, -3) \) and \( (-1, -5) \).
13Find the equation of the straight line through \( (0, -a) \) and \( (b, 0) \).
14Find the equation of the straight line through \( (p, 0) \) and \( (0, q) \).
15Find the equation of the straight line through \( (4, 3) \) and \( (5, 6) \).
16Prove that the points \( (1, -1) \), \( (5, 2) \) and \( (9, 5) \) are collinear.
17Prove that the points \( (-3, 0) \), \( (0, -6) \) and \( (-1, -4) \) are collinear.
18Prove that the points \( (2, 4) \), \( (4, 6) \) and \( (6, 8) \) are collinear.
19Prove that the points \( (1, -1) \), \( (2, 1) \) and \( (4, 5) \) are collinear.
20Prove that the points \( (7, -2) \), \( (2, 3) \) and \( (-1, 6) \) are collinear.
21Test whether the points \( (0, 4) \), \( (3, -2) \) and \( (6, 0) \) are collinear.
22The points \( A(b, 0) \), \( B(0, c) \) and \( C(3, 3) \) lie on the same straight line. Can B be the midpoint of AC? Give a reason.
23The points \( A(b, 0) \), \( B(0, c) \) and \( C(3, 3) \) lie on the same straight line. Using the two-point form, prove that \( \frac{1}{b} + \frac{1}{c} = \frac{1}{3} \).
24The points \( A(p, 0) \), \( B(0, q) \) and \( C(5, 5) \) lie on the same straight line. What does it mean to say the points are collinear?
25The points \( A(p, 0) \), \( B(0, q) \) and \( C(5, 5) \) lie on the same straight line. Using the two-point form, prove that \( \frac{1}{p} + \frac{1}{q} = \frac{1}{5} \).
26\( P(1, 0) \), \( Q(0, 1) \) and \( R(2, 3) \) are the vertices of a triangle. Draw triangle PQR.
27For the triangle with vertices \( P(1, 0) \), \( Q(0, 1) \) and \( R(2, 3) \), find the midpoint of side QR.
28For the triangle with vertices \( P(1, 0) \), \( Q(0, 1) \) and \( R(2, 3) \), find the equation of the median joining P to the midpoint of QR.
29\( A(2, 2) \), \( B(2, 8) \) and \( C(-6, 2) \) are the vertices of triangle ABC. Find the midpoint M of side BC.
30In triangle ABC with \( A(2, 2) \), \( B(2, 8) \) and \( C(-6, 2) \), find the length of the median AM.
31In triangle ABC with \( A(2, 2) \), \( B(2, 8) \) and \( C(-6, 2) \), prove that the equation of the median drawn from A is \( 3x + 4y = 14 \).
32\( X(-1, 3) \), \( Y(1, -1) \) and \( Z(5, 1) \) are the vertices of triangle XYZ. Find the length of the median XD drawn from X.
33For triangle XYZ with \( X(-1, 3) \), \( Y(1, -1) \) and \( Z(5, 1) \), find the equation of the median XD.
34The points \( A(a, b) \) and \( B(b, a) \) lie on the lines \( 6x - y = 1 \) and \( 2x - 5y = 5 \) respectively. Find the values of a and b.
35With \( a = 1 \) and \( b = 5 \), write the coordinates of the points \( A(a, b) \) and \( B(b, a) \).
36Find the distance between \( A(1, 5) \) and \( B(5, 1) \).
37Find the equation of AB, where \( A(1, 5) \) and \( B(5, 1) \).
38If A, B and C(x, y) lie on one line, do you get the same equation by equating the slope of AB with the slope of BC instead of the slope of CA? Think and discuss.
39When finding the line through two points, does it make a difference which point is chosen as \( (x_1, y_1) \) and which as \( (x_2, y_2) \)? Support your answer with an example.
40For the points \( A(-5, 3) \), \( B(-10, 6) \) and \( C(5, -3) \), is the equation of the line the same if you use A and C, or B and C, instead of A and B?
41Find the equation of the line passing through \( (-2, 5) \) with slope \( -3 \).
42Prove the point-slope form \( y - y_1 = m(x - x_1) \) starting from \( y = mx + c \).