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**Exercise 10.3**

**1. Reduce the following equations into slope-intercept form and find their slopes and the ****intercepts.**

**(i) **

**(ii) **

**(iii) **

**Ans. (i)** Given:

……….(i)

Comparing with we have and

**(ii)** Given:

……….(i)

Comparing with we have and

**(iii)** Given:

……….(i)

Comparing with we have and

**2. Reduce the following equations into intercept form and find their intercepts on the axis:**

**(i) **

**(ii) **

**(iii) **

**Ans. (i)** Given:

Comparing with , we have and

**(ii)** Given:

Comparing with , we have and

**(iii)** Given:

Comparing with , we have and

**3. Reduce the following equations into normal form. Find their perpendicular distances from the origin and angle between perpendicular ****axis:**

**(i) **

**(ii) **

**(iii) **

**Ans. (i)** Given:

Dividing both sides by we have

Putting and

=

Equation of line in normal form is

Comparing with we have and

**(ii)** Given:

Dividing both sides by we have

Putting and

Equation of line in normal form is

Comparing with we have and

**(iii)** Given:

Dividing both sides by we have

Putting and

Equation of line in normal form is

Comparing with we have and

**4. Find the distance of the point **** from the line **

**Ans. **Given: A line

Now, perpendicular distance of the point from the line is

= = = 5 units

**5. Find the points on the ****axis, where distances from the line **** are 4 units.**

**Ans. **Let the coordinates of the point on axis be

Now, perpendicular distance of the point from the line is

= =

According to question,

or

or

or

or

Therefore, the points on axis are (8, 0) and

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

**(i) **** and **

**(ii) **** and **

**Ans. (i)** Given: Two equations and

Here, and

Distance between two parallel lines =

= = = units

**(ii)** Given: Two equations and

Here, and

Distance between two parallel lines =

= = units

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

**Ans. **Given: Equation of a line which is parallel to the line is .

Since the line passes through point .

Therefore, the equation of required line is .

**8. Find the equation of the line perpendicular to the line **** and having **** intercept 3.**

**Ans. **Given: Equation of a line which is perpendicular to the line is .

Since the line passes through point .

Therefore, the equation of required line is .

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

**Ans. **Given:

Also

Let be the angle between the lines.

= = =

and

and

**10. The line through the points **** and (4, 1) intersects the line **** at right angle. Find the value of **

**Ans. **Slope of the line passing through the points and (4, 1) =

Also slope of the line is

Since both lines are perpendicular to each other.

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

**Ans. **Equation of the line parallel to the line is …..(i)

Since line (i) passes through , therefore …..(ii)

Subtracting eq. (ii) from eq. (i), we have

**12. Two lines passing through the point (2, 3) intersects each other at an angle of **** If slope of the line is 2, find equation of the other line.**

**Ans. **Given: and

Taking

Equation of required line is

Taking

Equation of required line is

**13. Find the equation of the right bisector of the line segment joining the points (3, 4) and **

**Ans. **Mid-point of the line segment joining the points 93, 4) and = = (1, 3)

Slope of the line joining points (3, 4) and =

Slope of the required line is

Therefore, the required line passes through point (1, 3) having slope

Equation of the required line

**14. Find the coordinates of the foot of perpendicular from the point **** on the line **

**Ans. **Let Q be the foot of perpendicular drawn from P on the line

Equation of a line perpendicular to is

Since the line passes through

Therefore, Q is a point of intersection A

of the lines and

Solving both the equations, we have and

Therefore, coordinates of foot of perpendicular are

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

**Ans. **Equation of the line PQ

Slope of the required line which is perpendicular to this line is

Equation of the line AB is

Comparing with we have and

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

**Ans. **Length of perpendicular from origin to line is

= =

And Length of perpendicular from origin to line is

=

= =

Now,

=

=

**17. In the triangle ABC with vertices A (2, 3), B**** and C (1, 2), find the equation and length of altitude from the vertex A.**

**Ans. **Slope of BC

Since AD BC, therefore slope of AD = 1

Equation of AD is

And Equation of BC is

Length of AD =

= units

**18. If **** is the length of perpendicular from the origin to the line whose intercepts on the axes are **** and **** then show that **

**Ans. **Given: Line

Now, is the length of perpendicular from origin to .

=

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