Find the distance between the points:

(i) A(2, -3) and B(-6, 3)

(ii) C(-1, -1) and D(8, 11)

(iii) P(-8, -3) and Q(-2, -5)

(iv) R(a + b, a – b) and S(a – b, a + b)

Find the distance of the point P(6, -6) from the origin.

If a point P(x, y) is equidistant from the points A(6, -1) and B(2, 3), find the relation between x and y.

Find a point on the x-axis which is equidistant from the points A(7, 6) and B(-3, 4).

Find the distance between the points A(x₁, y₁) and B(x₂, y₂), when

(i) AB is parallel to the x-axis

(ii) AB is parallel to the y-axis.

A is a point on the x-axis with abscissa -8 and B is a point on the y-axis with ordinate 15. Find the distance AB.

Find a point on the y-axis which is equidistant from A(-4, 3) and B(5, 2).

Using the distance formula, show that the points A(3, -2), B(5, 2) and C(8,8) are collinear.

Show that the points A(7, 10), B(-2, 5) and C(3, -4) are the vertices of an isosceles right-angled triangle.

Show that the points A(1, 1), B(-1, -1) and C(-√3, √3) are the vertices of an equilateral triangle each of whose sides is 22 units.

Show that the points A(2, -2), B(8, 4), C(5, 7) and D(-1, 1) are the angular points of a rectangle.

Show that A(3, 2), B(0, 5), C(-3, 2) and D(0, -1) are the vertices of a square.

Show that A(1, -2), B(3, 6), C(5, 10) and D(3, 2) are the vertices of a parallelogram.

Show that the points A(2, -1), B(3, 4), C(-2, 3) and D(-3, -2) are the vertices of a rhombus.

If the points A (-2, -1), B(1, 0), C(x, 3) and D(1, y) are the vertices of a parallelogram, find the values of x and y.

Find the area of ΔABC whose vertices are A(-3, -5), B(5, 2) and C(-9, -3).

Show that the points A(-5, 1), B(5, 5) and C(10, 7) are collinear.

Find the value of k for which the points A(-2, 3), B(1, 2) and C(k, 0) are collinear.

Find the area of the quadrilateral whose vertices are A(-4, 5), B(0, 7), C(5, -5) and D(-4, -2).

Find the area of ΔABC, the midpoints of whose sides AB, BC and CA are D(3, -1), E(5, 3) and F(1, -3) respectively.

Find the coordinates of the point which divides the join of A(-5, 11) and B(4, -7) in the ratio 2 : 7.

Find the ratio in which the x-axis cuts the join of the points A(4, 5) and B(-10, -2). Also, find the point of intersection.

In what ratio is the line segment joining the points A(-4, 2) and B(8, 3) divided by the y-axis? Also, find the point of intersection.

Find the slope of a line whose inclination is

(i) 30°

(ii) 120°

(iii) 135°

(iv) 90°

Find the inclination of a line whose slope is

(i)

Find the slope of a line which passes through the points

(i) (0, 0) and (4, -2)

(ii) (0, -3) and (2, 1)

(iii) (2, 5) and (-4, -4)

(iv) (-2, 3) and (4, -6)

If the slope of the line joining the points A(x, 2) and B(6, -8) is , find the value of x.

Show that the line through the points (5, 6) and (2, 3) is parallel to the line through the points (9, -2) and (6, -5)

Find the value of x so that the line through (3, x) and (2, 7) is parallel to the line through (-1, 4) and (0, 6).

Show that the line through the points (-2, 6) and (4, 8) is perpendicular to the line through the points (3, -3) and (5, -9).

If A(2, -5), B(-2, 5), C(x, 3) and D(1, 1) be four points such that AB and CD are perpendicular to each other, find the value of x.

Without using Pythagora’s theorem, show that the points A(1, 2), B(4, 5) and C(6, 3) are the vertices of a right-angled triangle.

Using slopes show that the points A(6, -1), B(5, 0) and C(2, 3) are collinear.

Using slopes, find the value of x for which the points A(5, 1), B(1, -1) and C(x, 4) are collinear.

Using slopes show that the points A(-4, -1), B(-2, -4), C(4, 0) and D(2, 3) taken in order, are the vertices of a rectangle.

Using slopes. Prove that the points A(-2, -1), B(1,0), C(4, 3) and D(1, 2) are the vertices of a parallelogram.

If the three points A(h, k), B(x₁, y₁) and C(x₂, y₂) lie on a line then show that (h – x₁)(y₂ – y₁) = (k – y₁)(x₂ – x).

If the points A(a, 0), B(0, b) and P(x, y) are collinear, using slopes, prove that

A line passes through the points A(4, -6) and B(-2, -5). Show that the line AB makes an obtuse angle with the x-axis.

The vertices of a quadrilateral are A(-4, -2), B(2, 6), C(8, 5) and D(9, -7). Using slopes, show that the midpoints of the sides of the quad. ABCD from a parallelogram.

Find the slope of the line which makes an angle of 30⁰ with the positive direction of the y-axis, measured anticlockwise.

Find the angle between the lines whose slopes are and .

Find the angle between the lines whose slopes are and

If A(1, 2), B(-3, 2) and C(3, 2) be the vertices of a ΔABC, show that

(i) tan A = 2

(ii)

(iii)

If θ is the angle between the lines joining the points (0, 0) and B(2, 3), and the points C(2, -2) and D(3, 5), show that .

If θ is the angle between the diagonals of a parallelogram ABCD whose vertices are A(0, 2), B(2,-1), C(4,

Show that the points A(0, 6), B(2, 1) and C(7, 3) are three corners of a square ABCD. Find (i) the slope of the diagonal BD and (ii) the coordinates of the fourth vertex D.

A(1, 1), B(7, 3) and C(3, 6) are the vertices of a ΔABC. If D is the midpoint of BC and AL ⊥ BC, find the slopes of (i) AD and (ii) AL.

Find the equation of a line parallel to the x - axis at a distance of

(i) 4 units above it

(ii) 5 units below it

Find the equation of a line parallel to the y - axis at a distance of

(i) 6 units to its right

(ii) 3 units to its left

Find the equation of a line parallel to the x - axis and having intercept - 3 on the y - axis.

Find the equation of a horizontal line passing through the point (4, - 2).

Find the equation of a vertical line passing through the point ( - 5, 6).

Find the equation of a line which is equidistant from the lines x = - 2 and x = 6.

Find the equation of a line which is equidistant from the lines y = 8 and y = - 2.

Find the equation of a line

whose slope is 4 and which passes through the point (5, - 7)

Find the equation of a line

whose slope is - 3 and which passes through the point ( - 2, 3);

Find the equation of a line

which makes an angle of with the positive direction of the x – axis and passes through the point (0, 2)

Find the equation of a line whose inclination with the x - axis is 30⁰ and which passes through the point (0, 5).

Find the equation of a line whose inclination with the x - axis is 150⁰ and which passes through the point (3, - 5).

Find the equation of a line passing through the origin and making an angle of 120⁰ with the positive direction of the x - axis.

Find the equation of a line which cuts off intercept 5 on the x - axis and makes an angle of 600 with the positive direction of the x - axis.

Find the equation of the line passing through the point P(4, - 5) and parallel to the line joining the points A(3, 7) and B( - 2, 4).

Find the equation of the line passing through the point P( - 3, 5) and perpendicular to the line passing through the points A(2, 5) and B( - 3, 6)

Find the slope and the equation of the line passing through the points:

(i) (3, - 2) and ( - 5, - 7)

Find the slope and the equation of the line passing through the points:

( - 1, 1) and (2, - 4)

Find the slope and the equation of the line passing through the points:

(5, 3) and ( - 5, - 3)

Find the slope and the equation of the line passing through the points:

(a, b) and ( - a, b)

Find the angle which the line joining the points and makes with the x - axis.

Prove that the points A(1, 4), B(3, - 2) and C(4, - 5) are collinear. Also, find the equation of the line on which these points lie.

If A(0, 0), b(2, 4) and C(6, 4) are the vertices of a ΔABC, find the equations of its sides.

If A ( - 1, 6), B( - 3, - 9) and C(5, - 8) are the vertices of a ΔABC, find the equations of its medians.

Find the equation of the perpendicular bisector of the line segment whose end points are A(10, 4) and B( - 4, 9).

Find the equations of the altitudes of a ΔABC, whose vertices are A(2, - 2), B(1, 1) and C( - 1, 0).

If A(4, 3), B(0, 0) and C(2, 3) are the vertices of a ΔABC, find the equation of the bisector of ∠A.

the midpoints of the sides BC, CA and AB of a ΔABC are D(2, 1), B( - 5, 7) and P( - 5, - 5) respectively. Find the equations of the sides of ΔABC.

If A(1, 4), B(2, - 3) and C( - 1, - 2) are the vertices of a ΔABC, find the equation of

(i) the median through A

(ii) the altitude through A

(iii) the perpendicular bisector of BC

Find the equation of the line whose

(i) slope = 3 and y - intercept = 5

(ii) slope = - 1 and y - intercept = 4

Find the equation of the line which makes an angle of 30⁰ with the positive direction of the x - axis and cuts off an intercept of 4 units with the negative direction of the y - axis.

Find the equation of the line whose inclination is and which makes an intercept of 6 units on the negative direction of the y - axis.

Find the equation of the line cutting off an intercept - 2 from the y - axis and equally inclined to the axes.

Find the equation of the bisectors of the angles between the coordinate axes.

Find the equation of the line through the point ( - 1, 5) and making an intercept of - 2 on the y - axis.

Find the equation of the line which is parallel to the line 2x – 3y = 8 and whose y - intercept is 5 units.

Find the equation of the line passing through the point (0, 3) and perpendicular to the line x – 2y + 5 = 0

Find the equation of the line passing through the point (2, 3) and perpendicular to the line 4x + 3y = 10

Find the equation of the line passing through the point (2, 4) and perpendicular to the x - axis.

Find the equation of the line that has x - intercept - 3 and which is perpendicular to the line 3x + 5y = 4

Find the equation of the line which is perpendicular to the line 3x + 2y = 8 and passes through the midpoint of the line joining the points (6, 4) and (4, - 2).

Find the equation of the line whose y - intercept is - 3 and which is perpendicular to the line joining the points ( - 2, 3) and (4, - 5).

Find the equation of the line passing through ( - 3, 5) and perpendicular to the line through the points (2, 5) and ( - 3, 6).

A line perpendicular to the line segment joining the points (1, 0) and (2, 3) divides it in the ratio 1 : 2. Find the equation of the line.

Find the equation of the line which cuts off intercepts -3 and 5 on the x-axis and y-axis respectively.

Find the equation of the line which cuts off intercepts 4 and -6 on the x-axis and y-axis respectively.

Find the equation of the line and cuts off equal intercepts on the coordinate axes and passes through the point (4,7).

Find the equation of the line which passes through the point (3, -5) and cuts off intercepts on the axes which are equal in magnitude but opposite in sign.

Find the equation of the line passing through the point (2, 2) and cutting off intercepts on the axes, whose sum is 9.

Find the equation of the line which passes through the point (22, -6) and whose intercept on the x-axis exceeds the intercept on the y-axis by 5.

Find the equation of the line whose portion intercepted between the axes is bisected at the point (3, -2).

Find the equation of the line whose portion intercepted between the coordinate axes is divided at the point (5, 6) in the ratio 3 : 1.

A straight line passes through the point (5, -2) and the portion of the line intercepted between the axes is divided at this point in the ratio 2 : 3. Find the equation of the line.

If the straight line = 1 passes through the points (8, -9) and (12, -15), find the values of a and b.

Find the equation of the line for which

p = 3 and ∝ = 450

Find the equation of the line for which

p = 5 and ∝ = 1350

Find the equation of the line for which

p = 8 and ∝ = 1500

Find the equation of the line for which

p = 3 and ∝ = 2250

Find the equation of the line for which

p = 2 and ∝ = 3000

Find the equation of the line for which

p = 4 and ∝ = 1800

The length of the perpendicular segment from the origin to a line is 2 units and the inclination of this perpendicular is ∝ such that sin ∝ = and ∝ is acute. Find the equation of the line.

Find the equation of the line which is at a distance of 3 units from the origin such that tan ∝ = where ∝ is the acute angle which this perpendicular makes with the positive direction of the x-axis.

Reduce the equation 2x – 3y – 5 = 0 to slope-intercept form, and find from it the slope and y-intercept.

Reduce the equation 5x + 7y – 35 = 0 to slope-intercept form, and hence find the slope and the y-intercept of the line

Reduce the equation y + 5 = 0 to slope-intercept form, and hence find the slope and the y-intercept of the line.

Reduce the equation 3x – 4y + 12 = 0 to intercepts form. Hence, find the length of the portion of the line intercepted between the axes

Reduce the equation 5x – 12y = 60 to intercepts form. Hence, find the length of the portion of the line intercepted between the axes

Find the inclination of the line:

(i) x + y + 6 =0

(ii) 3x + 3y + 8 = 0

Reduce the equation x + y - = 0 to the normal form x cos ∝ + y sin ∝ = p, and hence find the values of ∝ and p.

Reduce the equation to the normal form x cos ∝ + y sin ∝ = p, and hence find the values of ∝ and p.

Reduce each of the following equations to normal form :

(i) x + y - 2 = 0

(ii)

(iii) x + 5 = 0

(iv) 2y – 3 =0

(v) 4x + 3y - 9 = 0

Find the distance of the point (3, -5) from the line 3x – 4y = 27

Find the distance of the point (-2, 3) from the line 12x = 5y + 13.

Find the distance of the point (-4, 3) from the line 4(x + 5) = 3(y – 6).

Find the distance of the point (2, 3) from the line y = 4.

Find the distance of the point (4, 2) from the line joining the points (4, 1) and (2, 3)

Find the length of the perpendicular from the origin to each of the following lines :

(i) 7x + 24y = 50

(ii) 4x + 3y = 9

(iii) x = 4

Prove that the product of the lengths of perpendiculars drawn from the points

Find the values of k for which the length of the perpendicular from the point (4, 1) on the line 3x – 4y + k = 0 is 2 units

Show that the length of the perpendicular from the point (7, 0) to the line 5x + 12y – 9 = 0 is double the length of perpendicular to it from the point (2, 1)

The points A(2, 3), B(4, -1) and C(-1, 2) are the vertices of ΔABC. Find the length of the perpendicular from C on AB and hence find the area of ΔABC

What are the points on the x-axis whose perpendicular distance from the line is 4 units?

Find all the points on the line x + y = 4 that lie at a unit distance from the line 4x+3y=10.

A vertex of a square is at the origin and its one side lies along the line 3x – 4y – 10 = 0.

Find the area of the square.

Find the distance between the parallel lines 4x – 3y + 5 = 0 and 4x – 3y + 7 = 0

Find the distance between the parallel lines 8x + 15y – 36 = 0 and 8x + 15y + 32 = 0.

Find the distance between the parallel lines y = mx + c and y = mx + d

Find the distance between the parallel lines p(x + y) = q = 0 and p(x + y) – r =0

Prove that the line 12x – 5y – 3 = 0 is mid-parallel to the lines 12x – 5y + 7 = 0 and 12x – 5y – 13 = 0

The perpendicular distance of a line from the origin is 5 units, and its slope is -1. Find the equation of the line.

Find the points of intersection of the lines 4x + 3y = 5 and x = 2y – 7.

Show that the lines x + 7y = 23 and 5x + 2y = a 16 intersect at the point (2, 3).

Show that the lines 3x – 4y + 5 = 0, 7x – 8y + 5 = 0 and 4x + 5y = 45 are concurrent. Also find their point of intersection.

Find the value of k so that the lines 3x – y – 2 = 0, 5x + ky – 3 = 0 and 2x + y – 3 = 0 are concurrent.

Find the image of the point P(1, 2) in the line x – 3y + 4 = 0.

Find the area of the triangle formed by the lines x + y = 6, x – 3y = 2 and 5x – 3y + 2 = 0.

Find the area of the triangle formed by the lines x = 0, y = 1 and 2x + y = 2.

Find the area of the triangle, the equations of whose sides are y = x, y = 2x and y – 3x = 4.

Find the equation of the perpendicular drawn from the origin to the line 4x – 3y + 5 = 0. Also, find the coordinates of the foot of the perpendicular.

Find the equation of the perpendicular drawn from the point P(-2, 3) to the line x– 4y + 7 = 0. Also, find the coordinates of the foot of the perpendicular.

Find the equations of the medians of a triangle whose sides are given by the equations 3x + 2y + 6 = 0, 2x – 5y + 4 = 0 and x -3y – 6 = 0.

If the origin is shifted to the point (1, 2) by a translation of the axes, find the new coordinates of the point (3, -4).

If the origin is shifted to the point (-3, -2) by a translation of the axes, find the new coordinates of the point (3, -5).

If the origin is shifted to the point (0, -2) by a translation of the axes, the coordinates of a point become (3, 2). Find the original coordinates of the point.

If the origin is shifted to the point (2, -1) by a translation of the axes, the coordinates of a point become (-3, 5). Find the origin coordinates of the point.

At what point must the origin be shifted, if the coordinates of a point (-4,2) become (3, -2)?

Find what the given equation becomes when the origin is shifted to the point (1, 1).

x² + xy – 3x – y + 2 = 0

Find what the given equation becomes when the origin is shifted to the point (1, 1).

xy – y² – x + y = 0

Find what the given equation becomes when the origin is shifted to the point (1, 1).

x² – y² – 2x + 2y = 0

Find what the given equation becomes when the origin is shifted to the point (1, 1).

xy – x – y + 1 = 0

Transform the equation 2x² + y² – 4x + 4y = 0 to parallel axes when the origin is shifted to the point (1, -2).

Find the equation of the line drawn through the point of intersection of the lines x – 2y + 3 = 0 and 2x – 3y + 4 = 0 and passing through the point (4, -5).

Find the equation of the line drawn through the point of intersection of the lines x – y = 7 and 2x + y = 2 and passing through the origin.

Find the equation of the line drawn through the point of intersection of the lines x + y = 9 and 2x – 3y + 7 = 0 and whose slope is .

Find the equation of the line drawn through the point of intersection of the lines x – y = 1 and 2x – 3y + 1 = 0 and which is parallel to the line 3x + 4y = 12.

Find the equation of the line through the intersection of the lines 5x – 3y = 1 and 2x + 3y = 23 and which is perpendicular to the line 5x – 3y = 1.

Find the equation of the line through the intersection of the lines 2x – 3y = 0 and 4x – 5y = 2 and which is perpendicular to the line x + 2y + 1 = 0.

Find the equation of the line through the intersection of the lines x – 7y + 5 = 0 and 3x + y – 7 = 0 and which is parallel to x-axis.

Find the equation of the line through the intersection of the lines 2x – 3y + 1 = 0 and x + y – 2 = 0 and drawn parallel to y-axis.

Find the equation of the line through the intersection of the lines 2x + 3y – 2 = 0 and x – 2y + 1 = 0 and having x-intercept equal to 3.

Find the equation of the line passing through the intersection of the lines 3x – 4y + 1 = 0 and 5x + y – 1 = 0 and which cuts off equal intercepts from the axes.