Test Questions list on Geometric concept.

Question 1

Question:  

Answer:   

Line through the origin and perpendicular to x + y - 4 = 0 is
x - y = 0 ...(iv)
\ The foot of the perpendicular is given by solving (i) and (iv),
x = y = 2.
(2,2) are the coordinates of the foot of the perpendicular.
Line through the origin perpendicular to x + 5y - 26 =0 is
5x - y = 0 .......... (v)
Solving (ii) and (v) gives the coordinates of the foot of the perpendicular.
Coordinates of the foot of the perpendicular are (1,5).
Equation of the line perpendicular to 3x - y + 10 = 0 and through the origin is 3y + x = 0.
The foot of the perpendicular is obtained by solving the two equations 3x - y - 10 = 0 and x + 3y = 0.
We get x = 3, y = -1.
\ The coordinates of the foot of the perpendicular are (3,-1).
The three points are A(2,2), B(1, 5) and C(3, -1).


A,B and C are collinear.
Equation of the line joining the feet of the perpendicular is

Question 2

Question:   Show that the lines 3x - y - 6 = 0, 4x - y - 7 = 0 and 2x - y - 5 = 0 are concurrent.

Answer:    Given lines are

Solving (i) and (ii) for x and y, we get

The point of intersection is (1,-3).
Now substituting x = 1 and y = -3 in the LHS of (iii), we get

\ The three given lines are concurrent.

Question 3

Question:   Show that the point (3,4) and (-2,-5) lie on the opposite sides of the line 3x + 4y + 16 = 0.

Answer:    Let d = distance between (3,4) and the line 3x + 4y + 16 = 0.

d'= distance between (-2,-5) and the line 3x + 4y + 16 = 0.

Since d and d^| are of opposite signs, the points lie on either side of the line.

Question 4

Question:   Show that the origin and the point (2,-3) lie on the opposite sides of the line 6x - 5y + 8 = 0.

Answer:    Let d = distance between the origin and the line.

Let d' = distance between the point and the line.

Since d and d' are of opposite signs the origin and the point lie on opposite sides of the line.

Question 5

Question:   Find the equations of the bisectors of the angles between the lines 5x + 12y + 8 = 0 and 6x + 8y - 9 = 0.

Answer:    The given equations can be written as
5x + 12y + 8 = 0 and 6x + 8y - 9 = 0
Equation of the angle bisectors is



Taking only the positive sign, we get


Taking only the negative sign, we get

Question 6

Question:  

Answer:   





From (1) and (2) and by rule of cross multiplication, we get

Eliminating, we have

Question 7

Question:   Write down the equation of line with slope 4 and y-intercept 2.

Answer:    Since the slope and the intercept are given, the equation is of the form
y = mx+c.
Here m = 4 and c = 2
\The equation of the line is y = 4x + 2.

Question 8

Question:  
units on the negative direction of y-axis and makes an angle of 150^owith the positive direction of x-axis.

Answer:   
\ Required equation of the line is

or

Question 9

Question:  

Answer:    Let the equation of the line be







Aliter
Let P (x, y) be any point on the line, then

Question 10

Question:   Find the equation of the line passing through the points A (2,4) and B (-3,1).

Answer:    Let P (x, y) be any point on the line joining A and B.
Then,


Aliter:
Let equation of the line be
y = mx + c ..........(i)
(i) passes through (2,4),
\ 4 = 2m + c ......... (a)
(i) Passes through (-3,1),
\ 1 = -3m + c ......... (b)
Subtracting (b) from (a), we have



Now substituting these values in (i), we get