R.S. Aggarwal solutions for Mathematics [English] Class 10 ICSE chapter 14 - Equation of a Straight Line [Latest edition]

Find the slope of the line passing through the points: 

$$\mathrm{A}(-2, 1)$$ and $$\mathrm{B}(3, -4)$$

Find the slope of the line passing through the points: 

$$\mathrm{A}(0, -3)$$ and $$\mathrm{B}(2, 1)$$

Find the slope of the line passing through the points:

$$\mathrm{A}(4, -9)$$ and $$\mathrm{B}(-2, -1)$$

Find the slope of the line passing through the points:

$$\mathrm{A}(2, 5)$$ and $$\mathrm{B}(-4, -4)$$

If the slope of the line joining $$\mathrm{P}(k, 3)$$ and $$\mathrm{Q}(8, -6)$$ is $$\frac{-3}{4}$$, find the value of $$k$$.

Without using distance formula, prove that the points A(1, 4), B(3, –2) and C(–3, 16) are collinear.

Find the value of $$k$$ such that the points $$\mathrm{P}(k, 1), \mathrm{Q}(2, -5)$$ and $$\mathrm{R}(k -2, -3)$$ are collinear.

Find the equation of a line parallel to $$x$$-axis and passing through the point (–3, 2).

Find the equation of a line parallel to $$y$$-axis and passing through the point (–7, 5).

Find the equation of a line whose inclination is $$30^{\circ}$$ and whose $$y$$-intercept is –2.

Find the equation of a line whose slope $$= \frac{3}{4}$$ and $$y$$-intercept $$= -4$$.

Find the equation of a line whose gradient $$= \sqrt{3}$$ and $$y$$-intercept $$= \frac{-2}{3}$$.

Find the equation of a line which makes an angle of $$60^{\circ}$$ with the positive direction of $$x$$-axis and passes through the point $$\mathrm{P}(0, -3)$$.

Find the equation of a line whose slope is 4 and which passes through the point (3, 7).

Find the equation of a line whose slope is – 3 and which passes through the point (–2, 3).

Find the gradient and the $$y$$-intercept of the following lines:

$$5x - 10y = 3$$

Find the gradient and the $$y$$-intercept of the following lines:

$$\frac{x}{6} + \frac{y}{9} = 1$$

Find the gradient and the $$y$$-intercept of the following lines:

$$x + 4 = 0$$

Find the gradient and the $$y$$-intercept of the following lines:

$$y = 6$$

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

$$\mathrm{A}(-2, 1)$$ and $$\mathrm{B}(3, -4)$$

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

$$\mathrm{A}(4, -2)$$ and $$\mathrm{B}(2, -3)$$

In the given diagram, ABC is a triangle, where B(4, – 4) and C(– 4, –2). D is a point on AC.

1. Write down the coordinates of A and D.
2. Find the coordinates of the centroid of ΔABC.
3. If D divides AC in the ratio k : 1, find the value of k.
4. Find the equation of the line BD.

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

1. the equation of the line;
2. the co-ordinates of A and B;
3. the co-ordinates of M.

$$\mathrm{A}(2,3)$$ and $$\mathrm{B}(-2,5)$$ are two given points.
Find:

1. the gradient of AB;
2. the equation of AB;
3. the co-ordinates of the point, where AB intersects $$x$$-axis.

A straight line passes through the points A(2, – 4) and B(5, – 2).
Find:

1. the slope of the line AB;
2. the equation of the line AB;
3. the value of $$k$$, if AB passes through the point $$\mathrm{P}(k + 3, k - 4)$$.

If A(3, 4), B(7, –2) and C(–2, –1) are the vertices of a $$\triangle \mathrm{ABC}$$, write down the equation of the median through the vertex C.

A(2, 5), B(–1, 2) and C(5, 8) are the vertices of a triangle ABC, M is a point on AB such that AM : MB = 1 : 2. Find the coordinates of M. Hence find the equation of the line passing through the points C and M.

The vertices of a $$\triangle \mathrm{ABC}$$ are A(2, –11), B(2, 13) and C(–12, 1). Find the equations of its sides.

[Hint. The slope of AB = `(13 + 11)/(2 - 2) = 24/0`, which is not defined.
∴ AB is parallel to y-axis. Its equation is x = а.
Also, it passes through (2, –11). So, a = 2. Hence, x = 2.]

ABC is a triangle whose vertices are A(1, –1), B(0, 4) and C(–6, 4). D is the midpoint of BC. Find the co-ordinates of D.

ABC is a triangle whose vertices are A(1, –1), B(0, 4) and C(– 6, 4). D is the midpoint of BC. Find the equation of the median AD.

Find the equation of a line passing through the point $$(2, 3)$$ and intersecting the line $$2x - 3y = 6$$ on the $$y$$-axis.

[Hint. 2x – 3у = 6 mеets y-axis at (0, –2). So, the required line passes through (0, –2).]

Find the equation of a line with $$x$$-intercept $$= 5$$ and passing through the point $$(4, -3)$$.

[Hint. The line passes through the points A(5, 0) and B(4, –3).]

Find the equations of the diagonals of a rectangle whose sides are: $$x = -1, x = 4$$, $$y = -1$$ and $$y = 2$$.

[Hint. The equations of sides in order are x = –1, y = –1, x = 4, y = 2. So, its vertices are A(–1, –1), B(4, –1), C(4, 2) and D(–1, 2).]

Find the equation of the line passing through the point $$(3, 2)$$ and making positive equal intercepts on axes. Find the length of each intercept.

[Hint. Let each intercept be a units.
Then, this line passes through A(a, 0) and B(0, a).
Equation of AB is `(y - 0)/(x - a) = (a - 0)/(0 - a) ⇒ y = -x + a.`
Since, it passes through (3, 2), we get a = 5.
∴ Required equation is x + y = 5.]

Find the equation of the line passing through the origin and the point of intersection of the lines $$5x + 7y = 3$$ and $$2x - 3y = 7$$.

[Hint: On solving 5x + 7y = 3 and 2x – 3y = 7, we get x = 2, y = –1. Now, find the equation of a line passing through the points A(2, –1) and O(0, 0).]

M and N are two points on the X axis and Y axis respectively. P (3, 2) divides the line segment MN in the ratio 2 : 3.
Find:
(i) the coordinates of M and N
(ii) slope of the line MN.

In what ratio does the line $$x - 5y + 15 = 0$$ divide the join of $$\mathrm{A}(2,1)$$ and $$\mathrm{B}(-3, 6)$$? Also, find the co-ordinates of their point of intersection. 

[Hint. Let the given line divide AB in the ratio k : 1 at point P. Then, the co-ordinates of P would satisfy x – 5y + 15 = 0. Find k.]

Find the equations of the medians of $$\triangle \mathrm{ABC}$$ whose vertices are $$\mathrm{A}(-1, 2), \mathrm{B}(2, 1)$$ and $$\mathrm{C}(0, 4)$$. Hence, find the co-ordinates of the centroid of $$\triangle \mathrm{ABC}$$.

Find the coordinates of the centroid P of the ΔABC, whose vertices are A(–1, 3), B(3, –1) and C(0, 0). Hence, find the equation of a line passing through P and parallel to AB.

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

1. the co-ordinates of the fourth vertex D. 
2. length of diagonal BD. 
3. equation of side AB of the parallelogram ABCD.

In the given figure, ABC is a triangle and BC is parallel to $$y$$-axis. AB and AC intersect the $$y$$-axis at P and Q respectively.

1. Write the co-ordinates of A.
2. Find the length of AB and AC.
3. Find the ratio in which Q divides AC.
4. Find the equation of the line AC.

If $$\mathrm{A}(2, -3), \mathrm{B}(-5, 1), \mathrm{C}(7, -1)$$ and $$\mathrm{D}(0, k)$$ be four points such that AB is parallel to CD, find the value of $$k$$.

Show that the lines $$x + 2y - 5 = 0$$ and $$2x + 4y + 9 = 0$$ are parallel.

Find the value of $$k$$ for which the lines $$kx + 2y + 3 = 0$$ and $$8x + ky - 1 = 0$$ are parallel.

If the lines $$2x - by + 5 = 0$$ and $$ax + 3y = 2$$ are parallel, find the relation connecting $$a$$ and $$b$$.

Prove that the line through \(\mathrm{A}(-2, 6)\) and \(\mathrm{B}(4, 8)\) is perpendicular to the line through \(\mathrm{C}(8, 12)\) and \(\mathrm{D}(4, 24)\).

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

Prove that the lines 2x + 3y + 8 = 0 and 27x – 18y + 10 = 0 are perpendicular to each other.

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

If the straight lines 3x – 5y = 7 and 4x + ay + 9 = 0 are perpendicular to one another, find the value of a.

Without using Pythagoras Theorem, prove that the points \(\mathrm{A}(1, 3), \mathrm{B}(3, -1)\) and \(\mathrm{C}(-5, -5)\) are the vertices of a right-angled triangle.

[Hint: Show that \(\text{(Slope of AB)} \times (\text{slope of BC)} = -1\).]

Without using distance formula, show that the points A(1, –2), B(3, 6), C(5, 10) and D(3, 2) are the vertices of a parallelogram.

Given that A(5, 4), B(–3, –2) and C(1, –8) are the vertices of a $$\triangle \mathrm{ABC}$$.

Find: 

1. the slope of median AD,
2. the slope of altitude BM.

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

Find the equation of a line parallel to the line $$2x + y - 7 = 0$$ and passing through the intersection of the lines $$x + y - 4 = 0$$ and $$2x - y = 8$$.

A(-1, 3), B(4, 2) and C(3, -2) are the vertices of a triangle.

1) Find the coordinates of the centroid G of the triangle

2) Find the equation of the line through G and parallel to AC

Find the equation of a line passing through the point $$\mathrm{P}(-2, 1)$$ and parallel to the line joining the points $$\mathrm{A}(4, -3)$$ and $$\mathrm{B}(-1, 5)$$.

If the lines kx – y + 4 = 0 and 2y = 6x + 7 are perpendicular to each other, find the value of k.

Find the equation of a line parallel to 2y = 6x + 7 and passing through (−1, 1).

Find the equation of the line passing through the origin and perpendicular to the line $$y + 5x = 3$$.

Find the equation of a line passing through the origin and parallel to the line $$3x - 2y + 4 = 0$$.

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

[Hint: The required line passes through $$(-3, 0)$$.]

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

[Hint: Required line is parallel to $$y$$-axis.]

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

Find the equation of the line passing through the point of intersection of the lines $$5x - 8y + 23 = 0$$ and $$7x + 6y - 71 = 0$$ and perpendicular to the line $$4x - 2y = 3$$.

A line through origin meets the line $$2x = 3y + 13$$ at right angles at point Q. Find the co-ordinates of Q. 

[Hint. Find the co-ordinates of the foot of the perpendicular from $$P(0, 0)$$ to $$2x = 3y + 13$$.]

Find the equation of the perpendicular from the point $$\mathrm{P}(1, -2)$$ on the line $$4x - 3y - 5 = 0$$. Also, find the co-ordinates of the foot of the perpendicular.

Find the equation of the line which is perpendicular to the line $$\frac{x}{a} - \frac{y}{b} = 1$$ at the point where the given line meets $$y$$-axis.

Equation of a line AB is $$x + 2y + 6 = 0$$. A perpendicular PQ is dropped on AB from the point $$\mathrm{P}(3, -2)$$ meeting AB at Q. Find the: 

1. equation of PQ. 
2. coordinates of the point Q.

The points A(1, 3) and C(6, 8) are two opposite vertices of a square ABCD. Find the equation of the diagonal BD.

[Hint. $$BD$$ is the right-bisector of $$AC$$.]

A(1, 4), B(3, 2) and C(7, 5) are the vertices of a $$\triangle \mathrm{ABC}$$. 

Find:

1. the co-ordinates of the centroid G of $$\triangle \mathrm{ABC}$$, 
2. the equation of a line through G and parallel to AB.

A(– 4, 2), B(6, 4) and C(2, – 2) are the vertices of $$\triangle \mathrm{ABC}$$. Find the equation of median AD.

A(– 4, 2), B(6, 4) and C(2, – 2) are the vertices of $$\triangle \mathrm{ABC}$$. Find the equation of altitude BM.

A(– 4, 2), B(6, 4) and C(2, – 2) are the vertices of $$\triangle \mathrm{ABC}$$. Find the equation of right bisector of AB.

A(– 4, 2), B(6, 4) and C(2, – 2) are the vertices of $$\triangle \mathrm{ABC}$$. Find the co-ordinates of centroid of $$\triangle \mathrm{ABC}$$.

Find the equation of the perpendicular drawn from the point $$\mathrm{P}(2, 3)$$ on the line $$y = 3x + 4$$. Find the co-ordinates of the foot of the perpendicular. 

[Hint: Let $$AB$$ be the line $$y = 3x + 4$$ and let $$PQ \perp AB$$. 
Slope of $$PQ = \frac{-1}{\text {(Slope of AB)}} = \frac{-1}{3}$$. 
Equation of $$PQ$$ is $$\frac{y - 3}{x - 2} = \frac{-1}{3} \Rightarrow x + 3y = 11$$. 
On solving $$y = 3x + 4$$ and $$x + 3y = 11$$, we get co-ordinates of $$Q$$]

A(1, 2), B(2, 3) and C(4, 3) are the vertices of a $$\triangle \mathrm{ABC}$$.

Find:

1. the equation of altitude through B; 
2. the equation of altitude through C; 
3. the co-ordinates of the orthocentre of $$\triangle \mathrm{ABC}$$.

$$\mathrm{A}(1, 2), \mathrm{B}(3, -4)$$ and $$\mathrm{C}(5, -6)$$ are the vertices of $$\triangle \mathrm{ABC}$$. 

Find: 

1. the equation of the right bisector of BC; 
2. the equation of the right bisector of CA; 
3. the co-ordinates of the circumcentre of $$\triangle \mathrm{ABC}$$.

P is a point on the x-axis which divides the line joining A(–6, 2) and B(9, –4). Find:

1. the ratio in which P divides the line segment AB.
2. the coordinates of the point P.
3. equation of a line parallel to AB and passing through (–3, –2).

A line segment AB meets $$x$$-axis at A and $$y$$-axis at B. P(4, –1) divides AB in the ratio 1 : 2.

1. Find the co-ordinates of A and B. 
2. Find the equation of the line through P and perpendicular to AB.

The vertices of a $$\triangle \mathrm{ABC}$$ are A(3, 8), B(–1, 2) and C(6, –6).

Find:

1. Slope of BC.
2. Equation of a line perpendicular to BC and passing through A.

Line AB is perpendicular to CD. Co-ordinates of B, C and D are respectively (4, 0), (0, −1) and (4, 3). Find:

1. slope of CD
2. equation of AB

A and B are two points on the $$x$$-axis and $$y$$-axis respectively.

1. Write down the coordinates of A and B. 
2. P is a point on AB such that $$\mathrm{AP} : \mathrm{PB} = 3 : 1$$. Using section formula, find the coordinates of point P. 
3. Find the equation of a line passing through P and perpendicular to AB.

A line segment joining P(2, –3) and Q(0, –1) is cut by the x-axis at the point R. A line AB cuts the y-axis at T(0, 6) and is perpendicular to PQ at S.

Find the:

1. equation of line PQ
2. equation of line AB
3. coordinates of points R and S.

In the given graph ABCD is a parallelogram.

Using the graph, answer the following:

1. write down the coordinates of A, B, C and D.
2. calculate the coordinates of ‘P’, the point of intersection of the diagonals AC and BD.
3. find the slope of sides CB and DA and verify that they represent parallel lines.
4. find the equation of the diagonal AC.

The inclination of every line parallel to the x-axis is ______.

The inclination of y-axis is ______.

The slope of a line whose inclination is 30°, is ______.

The slope of a horizontal line is ______.

The slope of a vertical line is ______.

The slope of a line passing through two given points \(\mathrm{A}\left(x_{1}, y_{1}\right)\) and \(\mathrm{B}\left(x_{2}, y_{2}\right)\) is given by: