Coordinate Geometry

In an isosceles Δ 𝑷𝑸𝑹, PQ = PR. The base QR lies of x axis, P lies on the y –
axis and 2x -3y +9 = 0 is the equation of PQ. Find the equation of the straight line
along PR.

If the vertices of a Δ𝑨𝑩𝑪 𝒂𝒓𝒆 A (-4, 4), B (8, 4) and C(8, 10), then find the
equation of the straight line along the median from the vertex A.

I the vertices of a Δ𝑨𝑩𝑪 are A(2, -4), B(3, 3) and C(-1, 5), then find the
equation of the straight line along the altitude from the certex B.

Find the equation of the straight line joining the point of intersection of the
lines 3x – y +9 = 0 and x +2y =4 and the point of intersection of the lines
2x +y -4 = 0 and x- 2y +3 = 0.

Find the equation of the straight line which passes through the point of
intersection of the straight lines 5x – 6y =1 and 3x + 2y +5 = 0 and is
perpendicular to the straight line 3x -5y +11 = 0.

Find the equation of the straight line passing through the point of
intersection of the lines 2x + y -3 =0 and 5x + y -6 = 0 and parallel to the line
joining the points (1, 2) and (2, 1).

Find the equation of the perpendicular bisector of the straight line segment
joining the points (3, 4) and (-1, 2).

If the straight line passing through the points (h , 3) and (4, 1) intersects the
line 7x – 9y -19 =0 at right angle, then find the value of h.

Find the values of p for which the straight lines 8px + (2-3p)y +1 =0 and
Px + 8y -7=0 are perpendicular to each other.

Find the slope of the straight line
(i)3x +4y -6 =0, (ii) y =7x+6, (iii) 4x = 5y +3.

Find the equation of the line whose gradient is
𝟑
𝟐
and which passes through P,
where P divides the line segment joining A(-2, 6) and B( 3, -4) in the ratio 2:3
internally.

If A(3, 6) and C(-1, 2) are two vertices of a rhombus ABCD, then find the
equation of straight line that lies along the diagonal BD.

fine the equation of the line passing through (22, -6) and having intercept on
x-axis exceed the intercept on y=axis by 5.

A straight line cuts the coordinate axes at A and B. If the midpoint of AB is
(3, 2), then find the equation of AB.

Find the equation of the line passing through the point (9, -1) and having its
X –intercept thrice as its y – intercept.

Find the equation of the straight line passing through the point (5, -3) and
whose intercepts on the axes are equal in magnitude but opposite in sign.

Find the equation of the straight lines passing through the point (2, 2) and the
sum of the intercept is 9.

Find the equation of the straight line passing through the point (3, 4) and has
intercepts which are in the ratio 3:2.

By using the concept of the equation of the straight line, prove that the given
three points are collinear. (i) (4, 2), (7, 5) and (9, 7) (ii) (1,4), (3, -2) and (-3, 16).
Solution: (i) Let us find the equation of the line joining the points (4, 2), (7, 5).

Find the equation of the median from the vertex R in a Δ 𝑷𝑸𝑹 with vertices at
P (1, -3), Q(-2, 5) and R(-3, 4).

Write the equations of the straight lines parallel to x-axis which are at a distance
of 5 units from the x – axis.

Show that the opposites sides of a quadrilateral with vertices AA(-2, -4),
B (5, -1), C(6,4) and D(-1, -) taken in order are paralle

Using the concept of slope, show that the vertices (1, 2), (-2, 2), (-4, -3)
and (-1, -3) taken in order form a parallelogram.

The vertices of a Δ𝑨𝑩𝑪 are A (-5, 7), B(-4 , -5) and C (4, 5). Find the slopes of
the altitudes of the triangle.

A triangle has vertices at (6, 7), (2, -9) and (-4, 1). Find the slopes of its
medians.

The vertices of Δ 𝑨𝑩𝑪 𝒂𝒓𝒆 A (1, 8), B (-2, 4), C(8, -5). If M and N are the
midpoints of AB and AC respectively, find the slope of MN and hence verify that
MN is parallel to BC.

The line joining the points A (0, 5) and B(4, 2) is perpendicular to the line
joining the points C (-1, -2) and D(5, b). Find the value of b

The line joining the points A (-2 , 3) and B(a, 5) is parallel to the line joining
the points C (0, 5) and D (-2 , 1). Find the value of a.

Using the concept of slope, show that each of the following set points are
collinear.
(i) (2, 3) , (3, -1) and (4, -5) (ii) (4, 1), (-2, -3) and (-5, -5) (iii) (4, 4), (-2, 6) and
(1, 5)

Find the area of the triangle formed by joining the midpoints of the sides of a
triangle whose vertices are (0, -1), (2,1) and (0, 3). Find the ratio of this area to
the area of the triangle

If the three points (h, 0), (a, b) and (0, k) lie on a straight line, then using the
area of the triangle formula, show that
𝐚
𝐡
+
𝐛
𝐤
= 𝟏, where h, k ≠ 𝟎.

Find the area of then quadrilateral whose vertives are
(i) (6,9), (7,4), (4,2) and (3,7) (ii) (-3, 4) (-5, -6), (4,-1) and (1,2)
(iii) (-4, 5), (0, 7), (5, -5) AND (-4,-2)

Find the length of the medians of the triangle whose vertices are (1, -1, (0, 4)
and (-5, 3)

In what ratio is the line joining the points (-5, 1) and (2,3) divided by the y –
axis? Also, find the pointer of intersection.

Find the ratio in which the x – axis divides the line segment joining the points
(6, 4) and (1, -7).

Find the points which divide the line segment joining A (-4, 0) and B (0,6) into
four equal parts.

Find the points of trisection of the line segment joining the points A (2, -2) and
B (-7, 4).

Let A ( -6, -5) and B(-6, 4) be two points such that a point P on the line AB
satisfies AP =𝟐/𝟗
AB. Find the point P.

Find the coordinates of the foot of the perpendicular from the origin on the
straight line 3x +2y =3

If the vertices of a Δ 𝑨𝑩𝑪 𝒂𝒓𝒆 𝑨 𝟐, −𝟒 , B (3,3) and c(-1, 5), then find the
equation of the straight line along the altitude from the vertex B.

Find the equation of the straight line which passes through the point of
intersection of the straight line 3x – 5y +11 = 0

Find the equation of the perpendicular bisector of the straight lines segment
joining the points (3,4) and (-1, 2).

If the straight line passing through the points (h ,3) and (4,1) intersects the line
7x -9y 19 = 0 at right angle, then find the value of h.

Find the values of p for which the straight lines 8px + (-3p)y +1 =0 and
px+8y -7 are perpendicular to each other.

Find the value of a if the straight lines 5x -2y -9=0 and ay +2x-11 =0 are
perpendicular to each other.

Show that the straight lines 3x – 5y +7 =0 and 15x +9y +4 = 0 are
perpendicular

Show that the straight lines x + 2y +1 = 0 and 3x +6y +2 = 0 are parallel.

Find the slope of the straight line

Find the x and y – intercepts of the straight line
(i) 5x +3y -15 = 0 (ii) 2x –y + 16 = 0 (iii) 3x +3y =15.

Find equation of the straight line which passes through the midpoint of the
line segment joining (4, 2) and (3, 1) whose angle of inclination is 300.

Find the equation of the straight line whose (i) slope is -4 and passing through
(1,2) (ii) slope is 𝟐/𝟑
and passing through (5, -4).

Fine the slope and y – intercept of the line whose equation is (i) y = x+1
(ii) 5x =3y (iii) 4x -2y +1 =0, (iv) 10x + 15y +6=0.

Find the equation of the line intersecting the y-axis at a distance of 3 units
above the origin and tan 𝜽 =𝟏/𝟐
, where 𝜽 is the angle of inclination.

Find the equation of a straight line whose
(i) slope is -3 and y – intercept is 4,
(ii) angle of inclination is 600 and y – intercept is 3.

Find the equations of the straight lines parallel to the coordinate axes and
passing through the point (-5, -2)

Write the equation of the straight lines parallel to x – axis which are at a
distance of 5 units from the x – axis

The side AB of a sequence ABCD is parallel to x axis. Find the
(i) slope of AB (ii) slope of BC (iii) slope of the diagonal AC

Find the slope of the line which passes through the origin and the midpoint of
the line segment joining the points (0, -4) and (8, 0).

Find the slope of the straight line whose angle of inclination is
(i) 300 (ii) 600 (iii) 900

In each of the following, find the value of k for which the given points are
collinear
(i) (k, -1) , (2, 1) and (4, 5) (ii) (2, -5), (3, -4) and (9, k)
(iii) (k , k), (2,3) and (4, -1)

Determine if the following set of points are collinear or not
(i) (4, 3), (1, 2) and (-2 , 1) (ii) (-2 , -2), (-6, -2) and (-2, 2)
(iii) −𝟑/𝟐, 𝟑 , 𝟔, −𝟐 𝒂𝒏𝒅 −𝟑 , 𝟒

Determine if the following set of points are collinear or not
(i) (4, 3), (1, 2) and (-2 , 1) (ii) (-2 , -2), (-6, -2) and (-2, 2)
(iii) −
𝟑
𝟐
, 𝟑 , 𝟔, −𝟐 𝒂𝒏𝒅 −𝟑 , 𝟒

Vertices of the triangles taken in order and their areas are given below. In
each of the following find the value of a.
Vertices Area (in sq. units)
(i) (0, 0), (4, a), (6, 4) 17
(ii) (a, a), (4, 5), (6, 1) 09
(iii) (a, 3), (3, a), (-1,. 5) 12
Solution:

Find the area of the triangle formed by the points
(i) (0, 0), (3, 0) and (0, 2) (ii) (5, 2) (3, -5) and (-5, -1)
(iii) (-4, -5), (4, 5) and (-1, -6).

Let A (-6, -5) and B (-6, 4) be two points such that a point P on the line AB
satisfies AP =𝟐/𝟗AB. Find the point P.

Find the coordinates of the point which divides the line segment joining
(-3, 5) and (4, -9) in the ratio 1 : 6 internally.

find the coordinates of the point which divides the line segment joining
(-3,5) and (4,-9) in the r4atio 1: 6 internally.

If the centroid of a triangle is at (1, 3) and two of its vertices are (-7, 6) and
(8, 5), then find the third vertex of the triangle.

The centre of a circle is at (-6, 4). If one end of diameter of the circle is at the
origin, then find the other end.

Find the centroid of the triangle whose vertices are
(i) (1, 3), (2, 7) and (12, - 16) (ii) (3, -5), (-7, 4) and (10, -2)

Find the midpoint of the line segment joining the points.
(i) (1, - 1) and ( -5, 3) (ii) ( 0, 0) and (0, 4).

Find the midpoint of the line segment joining the points.

ABCD is a rectangle formed by the points A (- 1, - 1), B (- 1, 4), C (5, 4) and D (5, - 1). P, Q, R and S are the mid-points of AB, BC, CD, and DA respectively. Is the quadrilateral PQRS is a square? a rectangle? or a rhombus? Justify your answer.

Let A (4, 2), B (6, 5) and C (1, 4) be the vertices of ΔABC.

(i) The median from A meets BC at D. Find the coordinates of point D.

(ii) Find the coordinates of the point P on AD such that AP: PD = 2:1

(iii) Find the coordinates of point Q and R on medians BE and CF respectively such that BQ: QE = 2:1 and CR: RF = 2:1.

(iv) What do you observe?

(v) If A(x1, y1), B(x2, y2), and C(x3, y3) are the vertices of ΔABC, find the coordinates of the centroid of the triangle.

The vertices of a ΔABC are A (4, 6), B (1, 5) and C (7, 2). A line is drawn to intersect sides AB and AC at D and E respectively, such that. Calculate the area of the ΔADE and compare it with the area of ΔABC. (Recall Converse of basic proportionality theorem and Theorem 6.6 related to

ratio of areas of two similar triangles)

The class X students of a secondary school in Krishinagar have been allotted a rectangular plot of land for their gardening activity. Saplings of Gulmohar are planted on the boundary at a distance of 1 m from each other. There is a triangular grassy lawn in the plot as shown in the following figure. The students are to sow seeds of flowering plants on the remaining area of the plot.

(i) Taking A as origin, find the coordinates of the vertices of the triangle.

(ii) What will be the coordinates of the vertices of Δ PQR if C is the origin?

Also calculate the areas of the triangles in these cases. What do you observe?

The two opposite vertices of a square are (- 1, 2) and (3, 2). Find the coordinates of the other two vertices.

Find the centre of a circle passing through the points (6, - 6), (3, - 7) and (3, 3).

Find a relation between x and y if the points (x, y), (1, 2) and (7, 0) are collinear.

Determine the ratio in which the line 2x + y - 4 = 0 divides the line segment joining the points A(2, - 2) and B(3, 7)

You have studied in Class IX that a median of a triangle divides it into two triangles of equal areas. Verify this result for ΔABC whose vertices are A (4, - 6), B (3, - 2) and C (5, 2)

Find the area of the quadrilateral whose vertices, taken in order, are (- 4, - 2), (- 3, - 5), (3, - 2) and (2, 3)

Find the area of the triangle formed by joining the mid-points of the sides of the triangle whose vertices are (0, - 1), (2, 1) and (0, 3). Find the ratio of this area to the area of the given triangle.

In each of the following find the value of 'k', for which the points are collinear.

(i) (7, - 2), (5, 1), (3, - k) (ii) (8, 1), (k, - 4), (2, - 5)

Find the area of the triangle whose vertices are:

(i) (2, 3), (- 1, 0), (2, - 4) (ii) (- 5, - 1), (3, - 5), (5, 2)

Find the area of a rhombus if its vertices are (3, 0), (4, 5), ( - 1, 4) and ( - 2, - 1) taken in order. [Hint: Area of a rhombus = (product of its diagonals)]

Find the coordinates of the points which divide the line segment joining A (- 2, 2) and B (2, 8) into four equal parts.

If A and B are ( - 2, - 2) and (2, - 4), respectively, find the coordinates of P such that  and P lies on the line segment AB.

Find the coordinates of a point A, where AB is the diameter of circle whose centre is (2, - 3) and B is (1, 4)

If (1, 2), (4, y), (x, 6) and (3, 5) are the vertices of a parallelogram taken in order, find x and y.

Find the ratio in which the line segment joining A (1, - 5) and B (- 4, 5) is divided by the x-axis. Also find the coordinates of the point of division.

Find the ratio in which the line segment joining the points (- 3, 10) and (6, - 8) is divided by (- 1, 6).

To conduct Sports Day activities, in your rectangular shaped school ground ABCD, lines have been drawn with chalk powder at a distance of 1 m each. 100 flower pots have been placed at a distance of 1 m from each other along AD, as shown in the following figure. Niharika runs the distance AD on the 2nd line and posts a green flag. Preet runs the distance AD on the eighth line and posts a red flag. What is the distance between both the flags? If Rashmi has to post a blue flag exactly halfway between the line segment joining the two flags, where should she post her flag?

Find the coordinates of the points of trisection of the line segment joining (4, - 1) and (- 2, - 3).

Find the coordinates of the point which divides the join of (- 1, 7) and (4, - 3) in the ratio 2:3.

Find a relation between x and y such that the point (x, y) is equidistant from the point (3, 6) and (- 3, 4).

If Q (0, 1) is equidistant from P (5, - 3) and R (x, 6), find the values of x. Also find the distance QR and PR.

Find the values of y for which the distance between the points P (2, - 3) and Q (10, y) is 10 units.

Find the point on the x-axis which is equidistant from (2, - 5) and (- 2, 9).

Name the type of quadrilateral formed, if any, by the following points, and give reasons for your answer:

(i) (- 1, - 2), (1, 0), (- 1, 2), (- 3, 0)

(ii) (- 3, 5), (3, 1), (0, 3), (- 1, - 4)

(iii) (4, 5), (7, 6), (4, 3), (1, 2)

In a classroom, 4 friends are seated at the points A, B, C and D as shown in the following figure. Champa and Chameli walk into the class and after observing for a few minutes Champa asks Chameli, “Don't you think ABCD is a square?” Chameli disagrees.

Using distance formula, find which of them is correct.

Check whether (5, - 2), (6, 4) and (7, - 2) are the vertices of an isosceles triangle.

Determine if the points (1, 5), (2, 3) and (- 2, - 11) are collinear.

Find the distance between the points (0, 0) and (36, 15). Can you now find the distance between the two towns A and B discussed in Section 7.2.

Find the distance between the following pairs of points:

(i) (2, 3), (4, 1) (ii) (-5, 7), (-1, 3) (iii) (a, b), (- a, - b)