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**Miscellaneous Exercise**

**1. Find the values of K for which the line **** is:**

**(a) parallel to the axis **

**(b) parallel to the axis **

**(c) passing through the origin.**

**Ans. **Given: Equation of line

**(a) **If the line parallel to axis, then

**(b) **If the line parallel to axis, then

**(c)** If the line passes through origin then

K = 1 or K = 6

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

**Ans. **Given:

Dividing both sides by , we have

Putting and ,

Equation of line in normal form

Comparing with and

**3. Find the equations of the lines which cut-off intercepts on the axes whose sum and product are 1 and **** respectively.**

**Ans. **Let equation of line be and it is given that and

Now, solving and ,

Now, solving and

Therefore, the required line

And

**4. What are the points on the ****axis whose distance from the line **** is 4 units?**

**Ans. **Given: Equation of line and let point on axis be

Perpendicular distance from point to this line = =

According to question

Taking

Taking

Therefore, required points are and

**5. Find the perpendicular distance from the origin of the line joining the points **** and **

**Ans. **Equation of the line joining points and

Now, perpendicular distance from (0, 0) to this line,

=

=

=

=

=

=

**6. Find the equation of the line parallel to ****axis and drawn through the point of intersection of the lines **** and **

**Ans. **Solving and , we have and

Point of intersection of lines are and

But the required line is parallel to axis, therefore the equation of required line is .

**7. Find the equation of a line drawn perpendicular to the line **** through the point, where it meets the ****axis.**

**Ans. **Given: Equation of line

Slope of given line =

Since the required line is perpendicular to given line.

Slope of required line =

The given line meets the axis at (0, 6).

Equation of required line is

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

**Ans. **Given: Equations of lines are …..(i)

…..(ii)

And …..(iii)

On solving eq. (i) and (ii), we get the coordinates of point C (0, 0)

On solving eq. (ii) and (iii), we get the coordinates of point A

On solving eq. (i) and (iii), we get the coordinates of point B

Area of =

=

=

= sq. units

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

**Ans. **Given: Equations of lines are and

Three lines are concurrent if

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

**Ans. **Given: Equations of lines are and

Three lines are concurrent if

**11. Find the equations of the lines through the point (3, 2) which make an angle of **** with the line **

**Ans. **Let be the slope of required line which passes through point (3, 2),

Then the equation of required line is ……….(i)

The equation of given line

……….(ii)

Slope of given line =

According to question,

Taking

Then the equation of required line is

Taking

Then the equation of required line is

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

**Ans. **Given: Equations of lines and

Equation of any line through intersection of these lines

But, according to question

or

or

or

Equation of any line through intersection of these lines

Also Equation of any line through intersection of these lines

**13. Show that the equation of the line passing through the origin and making an angle **** with the line **** is **

**Ans. **Let be the slope of required line which passes through (0, 0).

Equation of line is

………(i)

Now, is the angle between and

or

or

or

or

Putting the value of in eq. (i), we get

**14. In what ratio, the line joining **** and (5, 7) is divided by the line **** ?**

**Ans. **Given: Equation of line

Let the given line divide the line joining A and B (5, 7) in the ratio at a point C.

Coordinates of C are

Since the point C lies on the given line.

Therefore, the required ratio is 1: 2.

**15. Find the distance of the line **** from the point (1, 2) along the line **

**Ans. **Here point of intersection of given line and are and

Distance between the points (1, 2) and

=

=

= units

**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 3 units from this point.**

**Ans. **Let the required line makes an angle with the positive direction of axis.

Equation of the line is

(given)

and

and

Since this point lies on

Squaring both side, we have

Therefore, the required line is parallel to axis or parallel to axis.

**17. The hypotenuse of a right angled triangle has its ends at the points (1, 3) and **** Find the equation of the legs (perpendicular sides) of the triangle.**

**Ans. **Let ABC be a right angled triangle with diagonal AC.

Slope of AC =

coordinate of A = coordinate of C = 1

AB = AC

A = C =

Let be the slope of lines with diagonal AC.

Taking

Taking

Equation of line AB and

and

and

Equation of line BC and

and

and

**18. Find the image of the point (3, 8) with respect to the line **** assuming the line to be a plane mirror.**

**Ans. **Let the image of the point A (3, 8) in the line mirror DE be C Then AC is perpendicular bisector of DE.

The coordinates of point B are

Since point B lies on the line

…..(i)

Since AC is perpendicular on DE.

Slope of AC x Slope of DE =

…..(ii)

Solving eq. (i) and (ii), we get and

Therefore, the image of point (3, 8) is

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

**Ans. **Let be the angle which the line makes with the line and

and

or

**20. If sum of the perpendicular distances of a variable point P**** from the lines **** and **** is always 10. Show that P must move on a line.**

**Ans. **Given: Equations of lines are …..(i)

and …..(ii)

Perpendicular distance of point P from line (i)

=

Perpendicular distance of point P from line (ii)

=

According to question,

When and

Then

, which represent a line.

Similarly, and and

and

In all the above cases, it represents a line therefore, P must move on a line.

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

**Ans. **The equations of parallel lines are and

Let A be any point which is equidistant from the parallel lines.

Taking

Therefore, the required line is

**22. A ray of light passing through the point A and the reflected ray passes through the point (1, 2) reflects on the ****axis at point A and the reflected ray passes through the point (5, 3). Find the coordinates of A.**

**Ans. **Let BA be the incident ray and AC be the reflected ray.

Now, for line AC

…..(i)

For line BA

…..(ii)

From eq. (i) and (ii),

Therefore, coordinates of point A are

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

**Ans. **Let and be the length of perpendiculars from and to the line

=

And

=

Now

=

=

=

=

= =

**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.**

**Ans. **The point of intersection of lines and if given by Since the shortest path through point A is perpendicular line AB.

Slope of line is

Slope of required line is

Therefore, the equation of required line is

Thanks for complete this math