## Selina Concise Mathematics Class 10 ICSE Solutions Equation of a Line

**Selina Publishers Concise Mathematics Class 10 ICSE Solutions Chapter 14 Equation of a Line**

### Equation of a Line Exercise 14A – Selina Concise Mathematics Class 10 ICSE Solutions

**Question 1.**

Find, which of the following points lie on the line x – 2y + 5 = 0:

(i) (1, 3) (ii) (0, 5)

(iii) (-5, 0) (iv) (5, 5)

(v) (2, -1.5) (vi) (-2, -1.5)

**Solution:**

**Question 2.**

**Solution:**

**Question 3.**

**Solution:**

**Question 4.**

For what value of k will the point (3, -k) lie on the line 9x + 4y = 3?

**Solution:**

The given equation of the line is 9x + 4y = 3.

Put x = 3 and y = -k, we have:

9(3) + 4(-k) = 3

27 – 4k = 3

4k = 27 – 3 = 24

k = 6

**Question 5.**

**Solution:**

**Question 6.**

Does the line 3x – 5y = 6 bisect the join of (5, -2) and (-1, 2)?

**Solution:**

**Question 7.**

(i) The line y = 3x – 2 bisects the join of (a, 3) and (2, -5), find the value of a.

(ii) The line x – 6y + 11 = 0 bisects the join of (8, -1) and (0, k). Find the value of k.

**Solution:**

**Question 8.**

(i) The point (-3, 2) lies on the line ax + 3y + 6 = 0, calculate the value of a.

(ii) The line y = mx + 8 contains the point (-4, 4), calculate the value of m.

**Solution:**

(i) Given, the point (-3, 2) lies on the line ax + 3y + 6 = 0.

Substituting x = -3 and y = 2 in the given equation, we have:

a(-3) + 3(2) + 6 = 0

-3a + 12 = 0

3a = 12

a = 4

(ii) Given, the line y = mx + 8 contains the point (-4, 4).

Substituting x = -4 and y = 4 in the given equation, we have:

4 = -4m + 8

4m = 4

m = 1

**Question 9.**

The point P divides the join of (2, 1) and (-3, 6) in the ratio 2: 3. Does P lie on the line x – 5y + 15 = 0?

**Solution:**

**Question 10.**

The line segment joining the points (5, -4) and (2, 2) is divided by the point Q in the ratio 1: 2. Does the line x – 2y = 0 contain Q?

**Solution:**

**Question 11.**

Find the point of intersection of the lines:

4x + 3y = 1 and 3x – y + 9 = 0. If this point lies on the line (2k – 1)x – 2y = 4; find the value of k.

**Solution:**

Consider the given equations:

4x + 3y = 1 ….(1)

3x – y + 9 = 0 ….(2)

Multiplying (2) with 3, we have:

9x – 3y = -27 ….(3)

Adding (1) and (3), we get,

13x = -26

x = -2

From (2), y = 3x + 9 = -6 + 9 = 3

Thus, the point of intersection of the given lines (1) and (2) is (-2, 3).

The point (-2, 3) lies on the line (2k – 1)x – 2y = 4.

(2k – 1)(-2) – 2(3) = 4

-4k + 2 – 6 = 4

-4k = 8

k = -2

**Question 12.**

Show that the lines 2x + 5y = 1, x – 3y = 6 and x + 5y + 2 = 0 are concurrent.

**Solution:**

We know that two or more lines are said to be concurrent if they intersect at a single point.

We first find the point of intersection of the first two lines.

2x + 5y = 1 ….(1)

x – 3y = 6 ….(2)

Multiplying (2) by 2, we get,

2x – 6y = 12 ….(3)

Subtracting (3) from (1), we get,

11y = -11

y = -1

From (2), x = 6 + 3y = 6 – 3 = 3

So, the point of intersection of the first two lines is (3, -1).

If this point lie on the third line, i.e., x + 5y + 2 = 0, then the given lines will be concurrent.

Substituting x = 3 and y = -1, we have:

L.H.S. = x + 5y + 2 = 3 + 5(-1) + 2 = 5 – 5 = 0 = R.H.S.

Thus, (3, -1) also lie on the third line.

Hence, the given lines are concurrent.

### Equation of a Line Exercise 14B – Selina Concise Mathematics Class 10 ICSE Solutions

**Question 1.**

**Solution:**

**Question 2.**

**Solution:**

**Question 3.**

Find the slope of the line passing through the following pairs of points:

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

(ii) (-4, 0) and origin

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

**Solution:**

**Question 4.**

Find the slope of the line parallel to AB if:

(i) A = (-2, 4) and B = (0, 6)

(ii) A = (0, -3) and B = (-2, 5)

**Solution:**

**Question 5.**

Find the slope of the line perpendicular to AB if:

(i) A = (0, -5) and B = (-2, 4)

(ii) A = (3, -2) and B = (-1, 2)

**Solution:**

**Question 6.**

The line passing through (0, 2) and (-3, -1) is parallel to the line passing through (-1, 5) and (4, a). Find a.

**Solution:**

**Question 7.**

The line passing through (-4, -2) and (2, -3) is perpendicular to the line passing through (a, 5) and (2, -1). Find a.

**Solution:**

**Question 8.**

Without using the distance formula, show that the points A (4, -2), B (-4, 4) and C (10, 6) are the vertices of a right-angled triangle.

**Solution:**

**Question 9.**

Without using the distance formula, show that the points A (4, 5), B (1, 2), C (4, 3) and D (7, 6) are the vertices of a parallelogram.

**Solution:**

**Question 10.**

(-2, 4), (4, 8), (10, 7) and (11, -5) are the vertices of a quadrilateral. Show that the quadrilateral, obtained on joining the mid-points of its sides, is a parallelogram.

**Solution:**

**Question 11.**

Show that the points P (a, b + c), Q (b, c + a) and R (c, a + b) are collinear.

**Solution:**

**Question 12.**

**Solution:**

**Question 13.**

**Solution:**

**Question 14.**

**Solution:**

**Question 15.**

A (5, 4), B (-3, -2) and C (1, -8) are the vertices of a triangle ABC. Find:

(i) the slope of the altitude of AB,

(ii) the slope of the median AD, and

(iii) the slope of the line parallel to AC.

**Solution:**

**Question 16.**

**Solution:**

**Question 17.**

**Solution:**

**Question 18.**

The points (-3, 2), (2, -1) and (a, 4) are collinear. Find a.

**Solution:**

**Question 19.**

The points (K, 3), (2, -4) and (-K + 1, -2) are collinear. Find K.

**Solution:**

**Question 20.**

Plot the points A (1, 1), B (4, 7) and C (4, 10) on a graph paper. Connect A and B, and also A and C.

Which segment appears to have the steeper slope, AB or AC?

Justify your conclusion by calculating the slopes of AB and AC.

**Solution:**

**Question 21.**

**Solution:**

### Equation of a Line Exercise 14C – Selina Concise Mathematics Class 10 ICSE Solutions

**Question 1.**

Find the equation of a line whose:

y-intercept = 2 and slope = 3.

**Solution:**

Given, y-intercept = c = 2 and slope = m = 3.

Substituting the values of c and m in the equation y = mx + c, we get,

y = 3x + 2, which is the required equation.

**Question 2.**

**Solution:**

**Question 3.**

**Solution:**

**Question 4.**

**Solution:**

**Question 5.**

Find the equation of the line passing through:

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

**Solution:**

**Question 6.**

The co-ordinates of two points P and Q are (2, 6) and (-3, 5) respectively. Find:

(i) the gradient of PQ;

(ii) the equation of PQ;

(iii) the co-ordinates of the point where PQ intersects the x-axis.

**Solution:**

**Question 7.**

The co-ordinates of two points A and B are (-3, 4) and (2, -1). Find:

(i) the equation of AB;

(ii) the co-ordinates of the point where the line AB intersects the y-axis.

**Solution:**

**Question 8.**

**Solution:**

**Question 9.**

In ΔABC, A = (3, 5), B = (7, 8) and C = (1, -10). Find the equation of the median through A.

**Solution:**

**Question 10.**

**Solution:**

**Question 11.**

Find the equation of the straight line passing through origin and the point of intersection of the lines x + 2y = 7 and x – y = 4.

**Solution:**

**Question 12.**

In triangle ABC, the co-ordinates of vertices A, B and C are (4, 7), (-2, 3) and (0, 1) respectively. Find the equation of median through vertex A.

Also, find the equation of the line through vertex B and parallel to AC.

**Solution:**

**Question 13.**

A, B and C have co-ordinates (0, 3), (4, 4) and (8, 0) respectively. Find the equation of the line through A and perpendicular to BC.

**Solution:**

**Question 14.**

Find the equation of the perpendicular dropped from the point (-1, 2) onto the line joining the points (1, 4) and (2, 3).

**Solution:**

**Question 15.**

Find the equation of the line, whose:

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

(ii) x-intercept = -4 and y-intercept = 6

(iii) x-intercept = -8 and y-intercept = -4

**Solution:**

**Question 16.**

Solution:

**Question 17.**

Find the equation of the line with x-intercept 5 and a point on it (-3, 2).

**Solution:**

**Question 18.**

Find the equation of the line through (1, 3) and making an intercept of 5 on the y-axis.

**Solution:**

**Question 19.**

Find the equations of the lines passing through point (-2, 0) and equally inclined to the co-ordinate axis.

**Solution:**

**Question 20.**

**Solution:**

**Question 21.**

**Solution:**

**Question 22.**

A (1, 4), B (3, 2) and C (7, 5) are vertices of a triangle ABC, Find:

(i) the co-ordinates of the centroid of triangle ABC.

(ii) the equation of a line, through the centroid and parallel to AB.

**Solution:**

**Question 23.**

A (7, -1), B (4, 1) and C (-3, 4) are the vertices of a triangle ABC. Find the equation of a line through the vertex B and the point P in AC; such that AP: CP = 2: 3.

**Solution:**

### Equation of a Line Exercise 14D – Selina Concise Mathematics Class 10 ICSE Solutions

**Question 1.**

Find the slope and y-intercept of the line:

(i) y = 4

(ii) ax – by = 0

(iii) 3x – 4y = 5

**Solution:**

**Question 2.**

The equation of a line x – y = 4. Find its slope and y-intercept. Also, find its inclination.

**Solution:**

**Question 3.**

(i) Is the line 3x + 4y + 7 = 0 perpendicular to the line 28x – 21y + 50 = 0?

(ii) Is the line x – 3y = 4 perpendicular to the line 3x – y = 7?

(iii) Is the line 3x + 2y = 5 parallel to the line x + 2y = 1?

(iv) Determine x so that the slope of the line through (1, 4) and (x, 2) is 2.

**Solution:**

**Question 4.**

**Solution:**

**Question 5.**

**Solution:**

**Question 6.**

(i) Lines 2x – by + 3 = 0 and ax + 3y = 2 are parallel to each other. Find the relation connecting a and b.

(ii) Lines mx + 3y + 7 = 0 and 5x – ny – 3 = 0 are perpendicular to each other. Find the relation connecting m and n.

**Solution:**

**Question 7.**

Find the value of p if the lines, whose equations are 2x – y + 5 = 0 and px + 3y = 4 are perpendicular to each other.

**Solution:**

**Question 8.**

The equation of a line AB is 2x – 2y + 3 = 0.

(i) Find the slope of the line AB.

(ii) Calculate the angle that the line AB makes with the positive direction of the x-axis.

**Solution:**

**Question 9.**

The lines represented by 4x + 3y = 9 and px – 6y + 3 = 0 are parallel. Find the value of p.

**Solution:**

**Question 10.**

If the lines y = 3x + 7 and 2y + px = 3 are perpendicular to each other, find the value of p.

**Solution:**

**Question 11.**

The line through A(-2,3) and B(4,b) is perpendicular to the line 2x – 4y =5. Find the value of b.

**Solution:**

**Question 12.**

Find the equation of the line through (-5, 7) and parallel to:

(i) x-axis (ii) y-axis

**Solution:**

**Question 13.**

(i) Find the equation of the line passing through (5, -3) and parallel to x – 3y = 4.

(ii) Find the equation of the line parallel to the line 3x + 2y = 8 and passing through the point (0, 1).

**Solution:**

**Question 14.**

Find the equation of the line passing through (-2, 1) and perpendicular to 4x + 5y = 6.

**Solution:**

**Question 15.**

Find the equation of the perpendicular bisector of the line segment obtained on joining the points (6, -3) and (0, 3).

**Solution:**

**Question 16.**

**Solution:**

**Question 17.**

B (-5, 6) and D (1, 4) are the vertices of rhombus ABCD. Find the equation of diagonal BD and of diagonal AC.

**Solution:**

**Question 18.**

A = (7, -2) and C = (-1, -6) are the vertices of square ABCD. Find the equations of diagonal BD and of diagonal AC.

**Solution:**

**Question 19.**

A (1, -5), B (2, 2) and C (-2, 4) are the vertices of triangle ABC, find the equation of:

(i) the median of the triangle through A.

(ii) the altitude of the triangle through B.

(iii) the line through C and parallel to AB.

**Solution:**

**Question 20.**

(i) Write down the equation of the line AB, through (3, 2) and perpendicular to the line 2y = 3x + 5.

(ii) AB meets the x-axis at A and the y-axis at B. Write down the co-ordinates of A and B. Calculate the area of triangle OAB, where O is the origin.

**Solution:**

**Question 21.**

The line 4x – 3y + 12 = 0 meets the x-axis at A. Write the co-ordinates of A.

Determine the equation of the line through A and perpendicular to 4x – 3y + 12 = 0.

**Solution:**

**Question 22.**

The point P is the foot of perpendicular from A (-5, 7) to the line whose equation is 2x – 3y + 18 = 0. Determine:

(i) the equation of the line AP

(ii) the co-ordinates of P

**Solution:**

**Question 23.**

The points A, B and C are (4, 0), (2, 2) and (0, 6) respectively. Find the equations of AB and BC.

If AB cuts the y-axis at P and BC cuts the x-axis at Q, find the co-ordinates of P and Q.

**Solution:**

**Question 24.**

**Solution:**

**Question 25.**

Find the value of a for which the points A(a, 3), B(2, 1) and C(5, a) are collinear. Hence, find the equation of the line.

**Solution:**

### Equation of a Line Exercise 14E – Selina Concise Mathematics Class 10 ICSE Solutions

**Question 1.**

Point P divides the line segment joining the points A (8, 0) and B (16, -8) in the ratio 3: 5. Find its co-ordinates of point P.

Also, find the equation of the line through P and parallel to 3x + 5y = 7.

**Solution:**

**Question 2.**

The line segment joining the points A(3, -4) and B (-2, 1) is divided in the ratio 1: 3 at point P in it. Find the co-ordinates of P. Also, find the equation of the line through P and perpendicular to the line 5x – 3y + 4 = 0.

**Solution:**

**Question 3.**

A line 5x + 3y + 15 = 0 meets y-axis at point P. Find the co-ordinates of point P. Find the equation of a line through P and perpendicular to x – 3y + 4 = 0.

**Solution:**

**Question 4.**

Find the value of k for which the lines kx – 5y + 4 = 0 and 5x – 2y + 5 = 0 are perpendicular to each other.

**Solution:**

**Question 5.**

**Solution:**

**Question 6.**

(1, 5) and (-3, -1) are the co-ordinates of vertices A and C respectively of rhombus ABCD. Find the equations of the diagonals AC and BD.

**Solution:**

**Question 7.**

Show that A (3, 2), B (6, -2) and C (2, -5) can be the vertices of a square.

(i) Find the co-ordinates of its fourth vertex D, if ABCD is a square.

(ii) Without using the co-ordinates of vertex D, find the equation of side AD of the square and also the equation of diagonal BD.

**Solution:**

**Question 8.**

A line through origin meets the line x = 3y + 2 at right angles at point X. Find the co-ordinates of X.

**Solution:**

**Question 9.**

A straight line passes through the point (3, 2) and the portion of this line, intercepted between the positive axes, is bisected at this point. Find the equation of the line.

**Solution:**

**Question 10.**

Find the equation of the line passing through the point of intersection of 7x + 6y = 71 and 5x – 8y = -23; and perpendicular to the line 4x – 2y = 1.

**Solution:**

**Question 11.**

**Solution:**

**Question 12.**

O (0, 0), A (3, 5) and B (-5, -3) are the vertices of triangle OAB. Find:

(i) the equation of median of triangle OAB through vertex O.

(ii) the equation of altitude of triangle OAB through vertex B.

**Solution:**

**Question 13.**

Determine whether the line through points (-2, 3) and (4, 1) is perpendicular to the line 3x = y + 1.

Does the line 3x = y + 1 bisect the line segment joining the two given points?

**Solution:**

**Question 14.**

**Solution:**

**Question 15.**

Find the value of k such that the line (k – 2)x + (k + 3)y – 5 = 0 is:

(i) perpendicular to the line 2x – y + 7 = 0

(ii) parallel to it.

**Solution:**

**Question 16.**

The vertices of a triangle ABC are A (0, 5), B (-1, -2) and C (11, 7). Write down the equation of BC. Find:

(i) the equation of line through A and perpendicular to BC.

(ii) the co-ordinates of the point, where the perpendicular through A, as obtained in (i), meets BC.

**Solution:**

**Question 17.**

**Solution:**

**Question 18.**

P (3, 4), Q (7, -2) and R (-2, -1) are the vertices of triangle PQR. Write down the equation of the median of the triangle through R.

**Solution:**

**Question 19.**

A (8, -6), B (-4, 2) and C (0, -10) are vertices of a triangle ABC. If P is the mid-point of AB and Q is the mid-point of AC, use co-ordinate geometry to show that PQ is parallel to BC. Give a special name of quadrilateral PBCQ.

**Solution:**

**Question 20.**

A line AB meets the x-axis at point A and y-axis at point B. The point P (-4, -2) divides the line segment AB internally such that AP : PB = 1 : 2. Find:

(i) the co-ordinates of A and B.

(ii) the equation of line through P and perpendicular to AB.

**Solution:**

**Question 21.**

A line intersects x-axis at point (-2, 0) and cuts off an intercept of 3 units from the positive side of y-axis. Find the equation of the line.

**Solution:**

**Question 22.**

Find the equation of a line passing through the point (2, 3) and having the x-intercept of 4 units.

**Solution:**

**Question 23.**

**Solution:**

**Question 24.**

**Solution:**

**Question 25.**

The ordinate of a point lying on the line joining the points (6, 4) and (7, -5) is -23. Find the co-ordinates of that point.

**Solution:**

**Question 26.**

Points A and B have coordinates (7, -3) and (1, 9) respectively. Find:

(i) the slope of AB.

(ii) the equation of the perpendicular bisector of the line segment AB.

(iii) the value of ‘p’ if (-2, p) lies on it.

**Solution:**

**Question 27.**

**Solution:**

**Question 28.**

The equation of a line 3x + 4y – 7 = 0. Find:

(i) the slope of the line.

(ii) the equation of a line perpendicular to the given line and passing through the intersection of the lines x – y + 2 = 0 and 3x + y – 10 = 0.

**Solution:**

**Question 29.**

ABCD is a parallelogram where A(x, y), B(5, 8), C(4, 7) and D(2, -4). Find:

(i) Co-ordinates of A

(ii) Equation of diagonal BD

**Solution:**

**Question 30.**

**Solution:**

**Question 31.**

**Solution:**

**Question 32.**

Find the equation of the line that has x-intercept = -3 and is perpendicular to 3x + 5y = 1.

**Solution:**

**Question 33.**

A straight line passes through the points P(-1, 4) and Q(5, -2). It intersects x-axis at point A and y-axis at point B. M is the mid- t point of the line segment AB. Find:

(i) the equation of the line.

(ii) the co-ordinates of points A and B.

(iii) the co-ordinates of point M

**Solution:**

**Question 34.**

**Solution:**

**Question 35.**

A line through point P(4, 3) meets x-axis at point A and the y-axis at point B. If BP is double of PA, find the equation of AB.

**Solution:**

**Question 36.**

Find the equation of line through the intersection of lines 2x – y = 1 and 3x + 2y = -9 and making an angle of 30° with positive direction of x-axis.

**Solution:**

**Question 37.**

Find the equation of the line through the Points A(-1, 3) and B(0, 2). Hence, show that the points A, B and C(1, 1) are collinear.

**Solution:**

**Question 38.**

Three vertices of a parallelogram ABCD taken in order are A(3, 6), B(5, 10) and C(3, 2), find :

(i) the co-ordinates of the fourth vertex D.

(ii) length of diagonal BD.

(iii) equation of side AB of the parallelogram ABCD.

**Solution:**

**Question 39.**

**Solution:**

**Question 40. **

**Question 41.**

i. Since A lies on the X-axis, let the co-ordinates of A be (x, 0).

Since B lies on the Y-axis, let the co-ordinates of B be (0, y).

Let m = 1 and n = 2

Using Section formula,

⇒ Slope of line perpendicular to AB = m = -2

P = (4, -1)

Thus, the required equation is

y – y_{1} = m(x – x_{1})

⇒ y – (-1) = -2(x – 4)

⇒ y + 1 = -2x + 8

⇒ 2x + y = 7

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