Problem 1 :
The slopes of the two lines are 7 and (3k + 2). If the two lines are parallel, find the value of k.
Problem 2 :
If the following equations of two lines are parallel, then find the value of k.
3x + 2y - 8 = 0
(5k + 3)x + 2y + 1 = 0
Problem 3 :
Find the equation of a straight line is passing through (2, 3) and parallel to the line 2x - y + 7 = 0.
Problem 4 :
Verify, whether the following equations of two lines are parallel.
3x + 2y - 7 = 0
y = -1.5x + 4
Problem 5 :
Verify, whether the following equations of two lines are parallel.
5x + 7y - 1 = 0
10x + 14y + 5 = 0
Problem 6 :
The slopes of the two lines are 7 and (3k + 2). If the two lines are perpendicular, find the value of k.
Problem 7 :
The equations of the two perpendicular lines are
3x + 2y - 8 = 0
(5k + 3) - 3y + 1 = 0
Find the value of k.
Problem 8 :
Find the equation of a straight line is passing through (2, 3) and perpendicular to the line 2x - y + 7 = 0.
Problem 9 :
Verify, whether the following two lines re perpendicular.
3x - 2y - 7 = 0
y = -(2x/3) + 4
Problem 10 :
Verify, whether the following two lines are perpendicular.
5x + 7y - 1 = 0
14x - 10y + 5 = 0
1. Answer :
If two lines are parallel, then their slopes are equal.
3k + 2 = 7
Subtract 2 from both sides.
3k = 5
Divide both sides by 5.
k = 5/3
2. Answer :
3x + 2y - 8 = 0
(5k + 3)x + 2y + 1 = 0
If the two lines are parallel, then their general forms of equations will differ only in the constant term and they will have the same coefficients of x and y.
To find the value of k, equate the coefficients of x.
5k + 3 = 3
Subtract 3 from both sides.
5k = 0
Divide both sides by 5.
k = 0
3. Answer :
Because the required line is parallel to 2x - y + 7 = 0, the equation of the required line and the equation of the given line 2x - y + 7 = 0 will differ only in the constant term.
Then, the equation of the required line is
2x - y + k = 0 ----(1)
The required line is passing through (2, 3).
Substitute x = 2 and y = 3 in (1).
2(2) - 3 + k = 0
4 - 3 + k = 0
1 + k = 0
k = -1
So, the equation of the required line is
(1)----> 2x - y - 1 = 0
4. Answer :
3x + 2y - 7 = 0
y = -1.5x + 4
In the equations of the given two lines, the equation of the second line is not in general form.
Let us write the equation of the second line in general form.
y = -1.5x + 4
1.5x + y - 4 = 0
Multiply by 2 on both sides,
3x + 2y - 8 = 0
Now, let us compare the equations of two lines,
3x + 2y - 7 = 0
3x + 2y - 8 = 0
The above two equations differ only in the constant term.
So, the equations of the given two lines are parallel.
5. Answer :
5x + 7y - 1 = 0
10x + 14y + 5 = 0
In the equation of the second line 10x + 14y + 5 = 0, the coefficients of x and y have the common divisor 2.
So, divide the second equation by 2.
5x + 7y + 2.5 = 0
Now, let us compare the equations of two lines,
5x + 7y - 1 = 0
5x + 7y + 2.5 = 0
The above two equations differ only in the constant term.
So, the equations of the given two lines are parallel.
6. Answer :
If the given two lines are perpendicular, then the product of the slopes is equal to -1.
7(3k + 2) = -1
Use distributive property.
21k + 14 = -1
Subtract 14 from each side.
21k = -15
Divide each side by 21.
k = -15/21
k = -5/7
7. Answer :
If the two lines are perpendicular, then the coefficient y term in the first line is equal to the coefficient of x term in the second line.
5k + 3 = 2
Subtract 3 from both sides.
5k = -1
Divide both sides by 5.
k = -1/5
8. Answer :
Required line is perpendicular to 2x - y + 7 = 0.
Then, the equation of the required line is
x + 2y + k = 0 ----(1)
The required line is passing through (2, 3).
Substitute x = 2 and y = 3 in (1).
(1)----> 2 + 2(3) + k = 0
2 + 6 + k = 0
8 + k = 0
Subtract 8 from both sides.
k = -8
9. Answer :
3x - 2y - 7 = 0
y = -(2x/3) + 4
In the equations of the given two lines, the equation of the second line is not in general form.
Let us write the equation of the second line in general form.
y = - (2x/3) + 4
Multiply each side by 3.
3y = - 2x + 12
2x + 3y - 12 = 0
Compare the equations of two lines,
3x - 2y - 7 = 0
2x + 3y - 12 = 0
When we look at the general form of equations of the above two lines, we get the following points.
(i) The sign of y terms are different.
(ii) The coefficient of x term in the first equation is the coefficient of y term in the second equation.
(iii) The coefficient of y term in the first equation is the coefficient of x term in the second equation.
(iv) The above equations differ in constant terms.
Considering the above points, it is clear that the given two lines are perpendicular.
10. Answer :
5x + 7y - 1 = 0
14x - 10y + 5 = 0
In the equation of the second line 14x - 10y + 5 = 0, the coefficients of 'x' and 'y' have the common divisor 2.
Divide the second equation by 2.
7x - 5y + 2.5 = 0
Compare the equations of two lines,
5x + 7y - 1 = 0
7x - 5y + 2.5 = 0
When we look at the general form of equations of the above two lines, we get the following points.
(i) The sign of y- terms are different.
(ii) The coefficient of x term in the first equation is the coefficient of y term in the second equation.
(iii) The coefficient of y term in the first equation is the coefficient of x term in the second equation.
(iv) The above equations differ in constant terms.
Considering the above points, it is clear that the given two lines are perpendicular.
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