**Q: 23 ** Prove that the product of the lengths of the perpendiculars drawn from the points and to the line is .

Given equation id line is
We can rewrite it as
Now, the distance of the line from the point is given by
Similarly,
The distance of the line from the point is given by
Hence proved

**Q : 24 **A person standing at the junction (crossing) of two straight paths represented by the equations and wants to reach the path whose equation is in the least time. Find equation of the path that he should follow.

point of intersection of lines and (junction) is
Now, person reaches to path in least time when it follow the path perpendicular to it
Now,
Slope of line is ,
let the slope of line perpendicular to it is , m
Then,
Now, equation of line passing through point and with slope is
Therefore, the required equation of line is

**Q : 22 ** A ray of light passing through the point reflects on the -axis at point and the reflected ray passes through the point . Find the coordinates of .

From the figure above we can say that
The slope of line AC
Therefore,
Similarly,
The slope of line AB
Therefore,
Now, from equation (i) and (ii) we will get
Therefore, the coordinates of . is

**Q : 21 ** Find equation of the line which is equidistant from parallel lines and .

Let's take the point which is equidistance from parallel lines and
Therefore,
It is that
Therefore,
Now, case (i)
Therefore, this case is not possible
Case (ii)
Therefore, the required equation of the line is

**Q: 20** If the sum of the perpendicular distances of a variable point from the lines and is always . Show that must move on a line.

Given the equation of line are
Now, perpendicular distances of a variable point from the lines are
Now, it is given that
Therefore,
Which is the equation of the line
Hence proved

**Q : 19 ** If the lines and are equally inclined to the line , find the value of .

Given equation of lines are
Now, it is given that line (i) and (ii) are equally inclined to the line (iii)
Slope of line is ,
Slope of line is ,
Slope of line is ,
Now, we know that
Now,
and
It is given that
Therefore,
Now, if
Then,
Which is not possible
Now, if
Then,
Therefore, the value of m is

**Q : 18 **Find the image of the point with respect to the line assuming the line to be a plane mirror.

Let point is the image of point w.r.t. to line
line is perpendicular bisector of line joining points and
Slope of line ,
Slope of line joining points and is ,
Now,
Point of intersection is the midpoint of line joining points and
Therefore,
Point of intersection is
Point also satisfy the line
Therefore,
On solving equation (i) and (ii) we will...

**Q : 17 ** The hypotenuse of a right angled triangle has its ends at the points and Find an equation of the legs (perpendicular sides) of the triangle.

Slope of line OA and OB are negative times inverse of each other
Slope of line OA is ,
Slope of line OB is ,
Now,
Now, for a given value of m we get these equations
If

**Q: 16 ** Find the direction in which a straight line must be drawn through the point so that its point of intersection with the line may be at a distance of units from this point.

Let be the point of intersection
it lies on line
Therefore,
Distance of point from is 3
Therefore,
Square both the sides and put value from equation (i)
When point is
and
When point is
Now, slope of line joining point and is
Therefore, line is parallel to x-axis -(i)
or
slope of line joining point and
Therefore, line is...

**Q : 15 ** Find the distance of the line from the point along the line .

point lies on line
Now, point of intersection of lines and is
Now, we know that the distance between two point is given by
Therefore, the distance of the line from the point along the line is

**Q : 14 ** In what ratio, the line joining and is divided by the line ?

Equation of line joining and is
Now, point of intersection of lines and is
Now, let's suppose point divides the line segment joining and in
Then,
Therefore, the line joining and is divided by the line in ratio

**Q : 13 ** Show that the equation of the line passing through the origin and making an angle

with the line is .

Slope of line is m
Let the slope of other line is m'
It is given that both the line makes an angle with each other
Therefore,
Now, equation of line passing through origin (0,0) and with slope is
Hence proved

**Q: 12 ** Find the equation of the line passing through the point of intersection of the lines and that has equal intercepts on the axes.

Point of intersection of the lines and is
We know that the intercept form of the line is
It is given that line make equal intercepts on x and y axis
Therefore,
a = b
Now, the equation reduces to
-(i)
It passes through point
Therefore,
Put the value of a in equation (i)
we will get
Therefore, equation of line is

**Q: 11 ** Find the equation of the lines through the point which make an angle of with the line .

Given the equation of the line is
The slope of line ,
Let the slope of the other line is,
Now, it is given that both the lines make an angle with each other
Therefore,
Now,
Case (i)
Equation of line passing through the point and with slope
Case (ii)
Equation of line passing through the point and with slope 3 is
Therefore,...

**Q : 10 ** If three lines whose equations are and are concurrent, then show that .

Concurrent lines means they all intersect at the same point
Now, given equation of lines are
Point of intersection of equation (i) and (ii)
Now, lines are concurrent which means point also satisfy equation (iii)
Therefore,
Hence proved

**Q : 9 ** Find the value of so that the three lines and may intersect at one point.

Point of intersection of lines and is
Now, must satisfy equation
Therefore,
Therefore, the value of p is

**Q : 8 ** Find the area of the triangle formed by the lines and .

Given equations of lines are
The point if intersection of (i) and (ii) is (0,0)
The point if intersection of (ii) and (iii) is (k,-k)
The point if intersection of (i) and (iii) is (k,k)
Therefore, the vertices of triangle formed by three lines are
Now, we know that area of triangle whose vertices are is
Therefore, area of triangle is

**Q : 7 ** Find the equation of a line drawn perpendicular to the line

through the point, where it meets the -axis.

given equation of line is
we can rewrite it as
Slope of line ,
Let the Slope of perpendicular line is m
Now, the ponit of intersection of and is
Equation of line passing through point and with slope is
Therefore, equation of line is

Q : 6 Find the equation of the line parallel to y-axis and drawn through the point of intersection of the lines and .

Point of intersection of the lines and
It is given that this line is parallel to y - axis i.e. which means their slopes are equal
Slope of is ,
Let the Slope of line passing through point is m
Then,
Now, equation of line passing through point and with slope is
Therefore, equation of line is

**Q : 5 ** Find perpendicular distance from the origin to the line joining the points and .

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