**Q : 4 **What are the points on the -axis whose distance from the line is units.

Given the equation of the line is
we can rewrite it as
Let's take point on y-axis is
It is given that the distance of the point from line is 4 units
Now,
In this problem
Case (i)
Therefore, the point is -(i)
Case (ii)
Therefore, the point is -(ii)
Therefore, points on the -axis whose distance from the line is units are and

**Q : 3 ** Find the equations of the lines, which cut-off intercepts on the axes whose sum and product are and , respectively.

Let the intercepts on x and y-axis are a and b respectively
It is given that
Now, when
and when
We know that the intercept form of the line is
Case (i) when a = 3 and b = -2
Case (ii) when a = -2 and b = 3
Therefore, equations of lines are

** Q : 2 ** Find the values of and , if the equation is the normal form of the line .

The normal form of the line is
Given the equation of lines is
First, we need to convert it into normal form. So, divide both the sides by
On comparing both
we will get
Therefore, the answer is

**Q : 1 ** Find the values of k for which the line is

(c) Passing through the origin.

Given equation of line is
It is given that it passes through origin (0,0)
Therefore,
Therefore, value of k is

**Q: 1 ** Find the values of for which the line is

(b) Parallel to the y-axis.

Given equation of line is
and equation of y-axis is
it is given that these two lines are parallel to each other
Therefore, their slopes are equal
Slope of is ,
and
Slope of line is ,
Now,
Therefore, value of k is

**Q : 1 ** Find the values of for which the line is

(a) Parallel to the x-axis.

Given equation of line is
and equation of x-axis is
it is given that these two lines are parallel to each other
Therefore, their slopes are equal
Slope of is ,
and
Slope of line is ,
Now,
Therefore, value of k is 3

**Q : 18 **If is the length of perpendicular from the origin to the line whose intercepts on

the axes are and , then show that .

we know that intercept form of line is
we know that
In this problem
On squaring both the sides
we will get
Hence proved

**Q : 17 ** In the triangle with vertices , and , find the equation and length of altitude from the vertex .

Let suppose foot of perpendicular is
We can say that line passing through point is perpendicular to line passing through point
Now,
Slope of line passing through point is ,
And
Slope of line passing through point is ,
lines are perpendicular
Therefore,
Now, equation of line passing through point (2 ,3) and slope with 1
-(i)
Now,...

**Q : 16 ** If and are the lengths of perpendiculars from the origin to the lines and , respectively, prove that .

Given equations of lines are and
We can rewrite the equation as
Now, we know that
In equation
Similarly,
in the equation
Now,
Hence proved

**Q : 15 ** The perpendicular from the origin to the line meets it at the point . Find the values of and .

We can say that line passing through point is perpendicular to line
Now,
The slope of the line passing through the point is ,
lines are perpendicular
Therefore,
- (i)
Now, the point also lies on the line
Therefore,
Therefore, the value of m and C is respectively

**Q : 14 ** Find the coordinates of the foot of perpendicular from the point to the line .

Let suppose the foot of perpendicular is
We can say that line passing through the point is perpendicular to the line
Now,
The slope of the line is ,
And
The slope of the line passing through the point is,
lines are perpendicular
Therefore,
Now, the point also lies on the line
Therefore,
On solving equation (i) and (ii)
we will get
Therefore,

**Q : 13 ** Find the equation of the right bisector of the line segment joining the points and .

Right bisector means perpendicular line which divides the line segment into two equal parts
Now, lines are perpendicular which means their slopes are negative times inverse of each other
Slope of line passing through points and is
Therefore, Slope of bisector line is
Now, let (h , k) be the point of intersection of two lines
It is given that point (h,k) divides the line segment...

**Q : 12 ** Two lines passing through the point intersects each other at an angle of . If slope of one line is , find equation of the other line.

Let the slope of two lines are respectively
It is given the lines intersects each other at an angle of and slope of the line is 2
Now,
Now, the equation of line passing through point (2 ,3) and with slope is
-(i)
Similarly,
Now , equation of line passing through point (2 ,3) and with slope is
-(ii)
Therefore,...

**Q : 11 **Prove that the line through the point and parallel to the line is

It is given that line is parallel to the line
Therefore, their slopes are equal
The slope of line ,
Let the slope of other line be m
Then,
Now, the equation of the line passing through the point and with slope is
Hence proved

**Q : 10 ** The line through the points and intersects the line at right angle. Find the value of .

Line passing through points ( h ,3) and (4 ,1)
Therefore,Slope of the line is
This line intersects the line at right angle
Therefore, the Slope of both the lines are negative times inverse of each other
Slope of line ,
Now,
Therefore, the value of h is

**Q : 9 ** Find angles between the lines and .

Given equation of lines are
and
Slope of line is,
And
Slope of line is ,
Now, if is the angle between the lines
Then,
Therefore, the angle between the lines is

**Q : 8 ** Find equation of the line perpendicular to the line and having intercept .

It is given that line is perpendicular to the line
we can rewrite it as
Slope of line ( m' ) =
Now,
The slope of the line is
Now, the equation of the line with intercept i.e. (3, 0) and with slope -7 is

**Q : 7 ** Find equation of the line parallel to the line and passing through the point .

It is given that line is parallel to line which implies that the slopes of both the lines are equal
we can rewrite it as
The slope of line =
Now, the equation of the line passing through the point and with slope is
Therefore, the equation of the line is

**Q : 6 ** Find the distance between parallel lines

(ii) and

Given equations of lines are
and
it is given that these lines are parallel
Therefore,
Now,
Therefore, the distance between two lines is

**Q : 6 ** Find the distance between parallel lines

(i) and .

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