Nootan solutions for मैथमैटिक्स [अंग्रेजी] कक्षा १० आईसीएसई chapter 12 - Equation of a line [Latest edition]

Find the slope of the line whose inclination is given as 60°.

Find the slope of the line whose inclination is given as 45°.

Find the inclination of a line whose slope (gradient) is 1.

Find the inclination of the line whose slope is `1/sqrt3`.

Find the equation of the line parallel to the Y-axis and at a distance of 3 units to the right of it.

Write the equation of the line: parallel to the Y-axis and at a distance of 5 units from it and to the left of it.

Find the equation of the line parallel to the X-axis and at a distance of 2 units above it.

Obtain the equation of the line which is: parallel to the X-axis and 3 units below it.

Find the equation of the line parallel to the X-axis and passing through the point (3, −4).

Find the equation of the line parallel to the X-axis and passing through the point (0, 2).

Find the equation of the line that is parallel to the Y-axis and passes through the point (3, −4).

Find the equation of the line perpendicular to the X-axis and passing through the origin.

Find the equation of the line perpendicular to the X-axis and passing through the point (−1, −1).

Find the equation of the line that passes through the point (−2, −5) and is perpendicular to the Y-axis.

Find the equation of a line through the origin that makes an angle with the positive X-axis equal to 45°.

Find the equation of the bisector of the angles between the coordinate axes in the first quadrant.

Determine the equation of a line passing through the point (−1, −2) and having slope `4/7`.

Determine the equation of a line passing through the point (−1, 2) with slope 4.

Find the equation of the line that satisfies the given condition:

Passing through the point (−4, 3) with slope `1/2`.

Determine the equation of a line passing through `(sqrt2, 2sqrt2) "with slope"  2/3`.

Determine the equation of a line passing through (2, 2) and inclined to the X-axis at an angle of 45°.

Determine the equation of a line passing through (−2, 3) and making an angle of 60° with the positive direction of the X-axis.

Find the equation of a line intersecting the X-axis at a distance of 3 units to the left of the origin and of slope −2.

Find the equation of the strainght line intersecting y-axis at a distance of 2 units above the origin and making an angle of 30° with the positive direction of the x-axis.

Find the equations of the line passing through the points (−1, 1) and (2, −4).

Find the equations of the line passing through the points (0, −3) and (5, 0).

Find the equations of the line passing through the points (−1, −2) and (2, 1).

Find the equations of the line passing through the points (2, 3) and (5, −2).

Find the equations of the median through vertex A of the triangle whose vertices are A(2, 5), B(−4, 9), and C(−2, −1).

Find the equations of the medians of a triangle, the coordinates of whose vertices are (−1, 6), (−3, −9) and (5, −8).

Show that the points (1, 4), (3, −2), and (−3, 16) are collinear. Find the equation of the line through them.

The vertices of a quadrilateral are A (−2, 6), B (1, 2), C (10, 4), and D (7, 8). Find the equation of its diagonals.

Find the equation of the diagonals of a rectangle whose sides are x = −1, x = 2, y = −2, and y = 6.

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

In what ratio is the line joining the points (2, 3) and (4, 1) divides the line segment joining the points (1, 2) and (4, 3)?

Find the equation of a line whose gradient is `-1/2` and which passes through a point M where M divides the line segment joining the points A(6, −2) and B(−4, 3) in the ratio 3 : 2.

Find the equation of a line with x-intercept 5 and passing through the point (−2, 3).

Find the equation of a line passing through the point (−3, 2) and the point of intersection of the lines x + y = 3 and x − 2y = 0.

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

A and B are two points on the X-axis and Y-axis, respectively. P(2, −1)

1. coordinates of A and B.
2. slope of AB.
3. equation of AB.

In the figure given, ABC is a triangle, and BC is parallel to the 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 radio in which Q divides AC. 
4. Find the equation of the line AC.

The line through P(5, 3) intersects the y-axis at Q.

1. Write the slope of the line.
2. Write the equation of the line.
3. Find the co-ordinates of Q.

A line passes through the point (3, 1) and cuts off positive intercepts on the X-axis and Y-axis in the ratio 2 : 3. Find the equation of the line.

Show that the line segment joining (2, 3) and (5, −7) is parallel to the line segment joining (2, −1) and (−1, 9).

Without using Pythagoras theorem, show that the points A(0, 4), B(1, 2) and C(−4, 2) are the vertices of a right-angled triangle.

If the lines `x/3 + y/4 = 7` and 3x + ky = 11 are perpendicular to each other, find the value of k.

Find the equation of a line, which is perpendicular to the line 2x − 4y + 12 = 0 and has a y-intercept of −3 units.

Find the equation of a straight line perpendicular to the line 3x − 4y + 12 = 0 and having the same y-intercept as 2x − y + 5 = 0.

Find the equation of the line which is parallel to 3x − 2y = −4 and passes through the point (0, 3).

Find the equation of the line passing through (0, 4) and parallel to the line 3x + 5y + 15 = 0.

The equation of a line is y = 3x − 5. Write down the slope of this line and the intercept made by it on the Y-axis. Hence, or otherwise, write down the equation of a line which is parallel to the line and which passes through the point (0, 5).

Write down the equation of the line perpendicular to 3x + 8y = 12 and passing through the point (−1, −2).

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.

Find the equation of the line that is parallel to 2x + 5y − 7 = 0 and passes through the mid-point of the line segment joining the points (2, 7) and (−4, 1).

Find the equation of the line that is perpendicular to 3x + 2y − 8 = 0 and passes through the mid-point of the line segment joining the points (5, −2) and (2, 2).

Find the equation of a straight line passing through the intersection of 2x + 5y − 4 = 0 with the X-axis and parallel to the line 3x − 7y + 8 = 0.

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

1. the slope of the line.
2. 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.

Find the equation of the perpendicular drawn from the point (1, −2) on the line 4x − 3у − 5 = 0. Also find the co-ordinates of the foot of the perpendicular.

Prove that the line through (0, 0) and (2, 3) is parallel to the line through (2, −2) and (6, 4).

Prove that the line through (−2, 6) and (4, 8) is perpendicular to the line through (8, 12) and (4, 24).

A(1, 4), B(3, 2) and C(7, 5) are vertices of a triangle ABC. Find the co-ordinates of the centroid of triangle ABC.

A(1, 4), B(3, 2) and C(7, 5) are vertices of a triangle ABC. Find the equation of a line through the centroid and parallel to AB.

А(2, −4), В(3, 3) and C(−1, 5) are the vertices of triangle ABC. Find the equation of the median of the triangle through A.

А(2, −4), В(3, 3) and C(−1, 5) are the vertices of triangle ABC. Find the equation of the altitude of the triangle through B.

Find the equation of the right bisector of the line segment joining the points (1, 2) and (5, 6).

Points A and B have co-ordinates (7, −3) and (1, 9) respectively. Find the slope of AB.

Points A and B have co-ordinates (7, −3) and (1, 9) respectively. Find the equation of the perpendicular bisector of the line segment AB, and the value of ‘p’ of (−2, p) lies on it.

The points B(1, 3) and D(6, 8) are two opposite vertices of a square ABCD. Find the equation of the diagonal AC.

ABCD is a rhombus. The co-ordinates of A and C are (3, 6) and (−1, 2) respectively. Write down the equation of BD.

Find the equation of the line passing through the intersection of the lines 4x + 3y = 1 and 5x + 4y = 2, and

1. parallel to the line x + 2y − 5 = 0 and
2. perpendicular to the X-axis.

Find the image of the point (1, 2) in the line x − 2y − 7 = 0.

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).

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.

If the lines px + 2y = 1 and x + 3y = 5 are parallel, then the value of p is ______.

If the lines 4x + 5y = 7 and x − ky + 1 = 0 are perpendicular, then the value of k is ______.

The y-intercept of the line 2x + 3y = 12 is ______.

If the line 3x − 5y = 15 meets the X-axis at point A, then the co-ordinates of A are ______.

Equation of a straight line parallel to x + y = 7, whose x-intercept is 3 units, is ______.

The equation of a line passing through (2, 1) and perpendicular to the line 3x − 4y = 8 is ______.

The straight lines 3x − 5y = 7 and 4x + ay + 9 = 0 are perpendicular to one another. The value of a is ______.

The slope of a line (6, k) and (1 − 3k, 3) is `1/2`. The value of k is ______.

Equation of a line whose x-intercept and y-intercept are 4 and 3 respectively, is ______.

Points A(x, y), B(3, −2) and C(4, −5) are collinear. The value of y in terms of x is ______.

1. Lines 2x + 3y = 5 and px + 6y = 1 are parallel then p = 1.
2. Equation of a line parallel to 2x + y = 4 and passing through (3, 5) is 2x + y = 2.

1. The straight lines kx + 2y = 1 and 4x - y = 3 are perpendicular then `k = 1/2`.
2. Two points form a line are y − y₁ = m(x − x₁).

1. The equation of a line with x-intercept 3 and y-intercept 2 is 2x + 3y = 12.
2. Two lines with slopes m₁ and m₂ are parallel if m₁ m₂ = −1.

1. The equation of a line parallel to the y-axis and passing through (3, 1) is y = 1.
2. The equation of a line with slope −3 and y-intercept 3 is 3x + y = 3.

If the lines 6x + 5y − 7 = 0 and 2px + 5y + 1 = 0 are parallel, find the value of p.

If the lines 3x + by + 5 = 0 and ax – 5y + 7 = 0 are perpendicular to each other, find the relation between a and b.

The co-ordinates of two points P and Q are (0, 4) and (3, 7), respectively. Find:

1. Equation of PQ.
2. Co-ordinates of the point where PQ intersects the X-axis.

Find the equation of a line passing through the point (2, −5) and making an intercept of −3 on the Y-axis.

Find the equation of the perpendicular bisector of the line segment joining the points (2, 3) and (6, −5).

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

The vertices of a Δ 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.

Using slopes, prove that the points (−1, −2), (5, 1) and (11, 4) are collinear.