Equation of Straight lines - Test Questions

Question 11

Question:   Find the equation of the straight line which passes through the origin and trisect the intercepts of the line 2x + 3y = 18 between the axes.

Answer:   
The given line can be written as


Intercepts on y-axis = b = 6 = OB
The points C and D are the points of trisection.


C = (6, 2)


D = (3, 4)

Question 12

Question:  
route from the place to canal is exactly northeast. A village is 5 miles north and 12 miles east from the place. Does the village lie by the nearer edge of the canal.

Answer:   



x + y = 17
The position of the village is (12, 5). The position of the village on the edge of the canal is 12 + 5 = 17, which is true.

Question 13

Question:  
i) Slope-intercept form
ii) Intercept form
iii) Normal form

Answer:    Given the line Ax + By + C = 0
i) Slope intercept form (y = mx + c)


Comparing this equation with y = mx+c, we have



...(i)








Multiplying by k throughout, we have
kAx + kBy = -kC







Substituting the value of k in (ii), we obtain
...(iii)


Note:
p is always positive.

When C > 0, equation (iii) will take the form

Again, when C < 0


Procedure to reduce the general equation to the normal form:
i) Put the constant term to the RHS and make it positive if not so, by changing the sign of every term.

Question 14

Question:  
i) Slope intercept form
ii) Intercept form
iii) Normal form

Answer:   





(iii) Normal form



Comparing with xcosa + ysina = p, we have



The equation in the normal form is

Question 15

Question:   Find the length of perpendicular drawn from (2,-1) on the line 3x + 4y = 12.

Answer:   
The length of the perpendicular drawn from (2,-1) on the line
3x + 4y = 12 is

Question 16

Question:   If P and P' are the length of the perpendiculars from the origin respectively, prove that 4p² + p² = a².

Answer:    The two lines are





4p² = a²sin²2q ....(iv)

p'² = a²cos²2q ....(v)
Adding (iv) and (v), we get
4p²+ p'² = a²

Question 17

Question:   Find the distance between the pair of parallel lines
3x + 4y - 15 = 0 and 6x + 8y = 25.

Answer:    The given lines are 3x + 4y - 15 = 0 ... (i)
6x + 8y - 25 = 0 ... (ii)
The perpendicular distance of 3x + 4y - 15 = 0 from the origin is

The perpendicular distance of 6x + 8y - 25 = 0 from the origin is

Question 18

Question:   Find the point of intersection of the lines.

Answer:    a)




b)

Question 19

Question:   Find the coordinates of the foot of the perpendicular from (4, - 4) on the line 5x + 7y - 12 = 0.

Answer:   
The equation of the line perpendicular to 5x + 7y - 12 = 0 is
7x - 5y + k = 0.
The line passes through (4,-4).

The line is 7x - 5y - 48 = 0.
To find the foot of the perpendicular, solve the equations

Question 20

Question:  

Answer:    The given lines are

The coordinates of the point of intersection of the lines (i) and (ii) are

The coordinates of the point of intersection of (i) and (ii) are (1,1).


Hence the three lines are concurrent.

Aliter:
The determinant



\The three lines are concurrent.