Note: If and intersect at a point , then and

Question 1: Find the point of intersection of the following pairs of lines :

i) and ii) and

iii) and

Answer:

i) Given and

Here comparing with we get

Similarly, comparing with we get:

Therefore,

Hence the coordinates of the intersection of the lines and are

ii) Given and

Here comparing with we get

Similarly, comparing with we get:

Therefore,

Hence the coordinates of the intersection of the lines and are

iii) Given and

Here comparing with we get

Similarly, comparing with we get:

Therefore,

Hence the coordinates of the intersection of the lines and are

Question 2: Find the coordinates of the vertices of a triangle, the equations of whose sides are:

i) and

ii) and

Answer:

i) Given equations:

… … … … … i)

… … … … … ii)

… … … … … iii)

First consider equations i) and ii):

Here comparing with we get

Similarly, comparing with we get:

Therefore,

Hence the coordinates of the intersection of the lines and are

Now consider equations ii) and iii):

Here comparing with we get

Similarly, comparing with we get:

Therefore,

Hence the coordinates of the intersection of the lines and are

Now consider equations iii) and i):

Here comparing with we get

Similarly, comparing with we get:

Therefore,

Hence the coordinates of the intersection of the lines and are

Therefore the three vertices of the triangle are , ,

ii) Given equations:

… … … … … i)

… … … … … ii)

… … … … … iii)

First consider equations i) and ii):

Here comparing with we get

Similarly, comparing with we get:

Therefore,

Hence the coordinates of the intersection of the lines and are

Now consider equations ii) and iii):

Here comparing with we get

Similarly, comparing with we get:

Therefore,

Hence the coordinates of the intersection of the lines and are

Now consider equations iii) and i):

Here comparing with we get

Similarly, comparing with we get:

Therefore,

Hence the coordinates of the intersection of the lines and are

Therefore the three vertices of the triangle are , ,

Question 3: Find the area of the triangle formed by the lines

i) , and

ii) and

iii) and

Answer:

i) Given equations:

… … … … … i)

… … … … … ii)

… … … … … iii)

First consider equations i) and ii):

Here comparing with we get

Similarly, comparing with we get:

Therefore,

Hence the coordinates of the intersection of the lines and are

Now consider equations ii) and iii):

Here comparing with we get

Similarly, comparing with we get:

Therefore,

Hence the coordinates of the intersection of the lines and are

Now consider equations iii) and i):

Here comparing with we get

Similarly, comparing with we get:

Therefore,

Hence the coordinates of the intersection of the lines and are

Therefore the three vertices of the triangle are , ,

Area of triangle by the above vertices

ii) Given equations:

… … … … … i)

… … … … … ii)

… … … … … iii)

First consider equations i) and ii):

Here comparing with we get

Similarly, comparing with we get:

Therefore,

Hence the coordinates of the intersection of the lines and are

Now consider equations ii) and iii):

Here comparing with we get

Similarly, comparing with we get:

Therefore,

Hence the coordinates of the intersection of the lines and are

Now consider equations iii) and i):

Here comparing with we get

Similarly, comparing with we get:

Therefore,

Hence the coordinates of the intersection of the lines and are

Therefore the three vertices of the triangle are

Area of triangle by the above vertices

sq. units

iii) Given equations:

… … … … … i)

… … … … … ii)

… … … … … iii)

First consider equations i) and ii):

Here comparing with we get

Similarly, comparing with we get:

Therefore,

Hence the coordinates of the intersection of the lines and are

Now consider equations ii) and iii):

Here comparing with we get

Similarly, comparing with we get:

Therefore,

Hence the coordinates of the intersection of the lines and are

Now consider equations iii) and i):

Here comparing with we get

Similarly, comparing with we get:

Therefore,

Hence the coordinates of the intersection of the lines and are

Therefore the three vertices of the triangle are

Area of triangle by the above vertices

sq. units

Question 4: Find the equations of the medians of a triangle, the equations of whose sides are: , and

Answer:

Given equations:

… … … … … i)

… … … … … ii)

… … … … … iii)

Solving i) and ii) we get

Solving ii) and iii) we get

Solving iii) and i) we get

Midpoint of

Therefore the equation of median :

Midpoint of

Therefore the equation of median :

Midpoint of

Therefore the equation of median :

Hence the equations of the three medians are , and

Question 5: Prove that the lines and form an equilateral triangle.

Answer:

Given equations:

… … … … … i)

… … … … … ii)

… … … … … iii)

Solving i) and ii) we get

Solving ii) and iii) we get

Solving iii) and i) we get

Length of units

Length of units

Length of units

Since

Therefore the is an equilateral triangle.

Question 6: Classify the following pairs of lines as co-incident, parallel or intersecting:

i) and ii) and

iii) and

Answer:

i) and

We will transform the equation into the form . Therefore :

Since , the lines are intersecting.

ii) and

We will transform the equation into the form . Therefore :

Since , and the intercepts are different, therefore the lines are parallel.

iii) and

We will transform the equation into the form . Therefore :

Since , the lines are parallel. Their intercepts are also the same. Hence we can say that the lines are coincident line.

Question 7: Find the equation of the line joining the point to the point of intersection of the lines and

Answer:

Given and

Here comparing with we get

Similarly, comparing with we get:

Therefore,

Therefore the equation of the line passing through and is:

Question 8: Find the equation of a line passing through the point of intersection of the lines and that has equal intercept on x-axis.

Answer:

Given and

Here comparing with we get

Similarly, comparing with we get:

Therefore,

Therefore

If the intercepts of the line are equal, then the equation of the line is:

This line passes through , therefore

Therefore equation of line is

Question 9: Show that the area of the triangle formed by the lines and is equal to , where and are roots of the equation

Answer:

Given equations: and

The vertices of the triangle are

Area of a triangle

Now are the roots of , therefore

Substituting we get

Area of triangle

. Hence proved.

Question 10: If the straight line passes through the intersection of the lines , and and is parallel to , find and .

Answer:

Given and

Here comparing with we get

Similarly, comparing with we get:

Therefore,

Therefore

If passes through , then

… … … … … i)

Since is parallel to , the slope of both the lines is the same.

… … … … … ii)

Substituting ii) in i) we get

Question 11: Find the orthocenter of the triangle the equations of whose sides are and

Answer:

Given equations:

… … … … … i)

… … … … … ii)

… … … … … iii)

Solving i) and ii) we get

Solving i) and iii) we get

Slope of Therefore Slope of

Therefore equation of :

… … … … … iv)

Slope of Therefore slope of

Equation of :

… … … … … v)

Now solving equation iv) and v) i.e. and

Here comparing with we get

Similarly, comparing with we get:

Therefore,

Therefore

Question 12: Three sides and of a triangle are and respectively. Find the equation of the altitude through the vertex .

Answer:

Given and

Here comparing with we get

Similarly, comparing with we get:

Therefore,

Therefore point of intersection

Slope of

Therefore Slope of

Therefore the equation of :

Question 13: Find the coordinates of the orthocenter of the, triangle whose vertices are and .

Answer:

Let and

Slope of

Therefore Slope of

Therefore equation of :

… … … … … i)

Slope of

Therefore Slope of

Therefore equation of :

… … … … … ii)

Now solve i) and ii) and

Here comparing with we get

Similarly, comparing with we get:

Therefore,

Therefore the orthocenter is

Question 14: Find the coordinates of the incentre and centroid of the triangle whose sides have the equations and .

Answer:

Given:

… … … … … i)

… … … … … ii)

… … … … … iii)

Solving i) and ii) i.e. and

Here comparing with we get

Similarly, comparing with we get:

Therefore,

Therefore the point of intersection is

Solving ii) and iii) i.e. and

Here comparing with we get

Similarly, comparing with we get:

Therefore,

Therefore the point of intersection is

Solving iii) and i) i.e. and

Here comparing with we get

Similarly, comparing with we get:

Therefore,

Therefore the point of intersection is

Now we need to find the lengths of the lines and .

Hence

Also

Therefore Centroid

Incenter

Question 15: Prove that the lines and form a rhombus.

Answer:

Given:

… … … … … i)

… … … … … ii)

… … … … … iii)

… … … … … iv)

Solving i) and ii) we get

Solving ii) and iii) we get

Solving iii) and iv) we get

Solving iv) and v) we get

Now let’s find the lengths of the sides:

Since the four lines are equal, ABCD is a rhombus.

Question 16: Find the equation of the line passing through the intersection of the lines and and parallel to line .

Answer:

Given lines: and

Solving the above lines gives us the point of intersection as

Slope of line is

Therefore the slope of the required line is as they are parallel

Therefore the equation of the required line:

Question 17: Find the equation of the straight line passing through the intersection of the lines and and perpendicular to line .

Answer:

Given lines: and

Solving the above equations, we get the point of intersection .

Slope of the line is

Therefore the slope of the line perpendicular to line is

Therefore the equation of the required line: