NCERT Solutions class-11 Maths Exercise 10.3

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

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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

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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

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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

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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

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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

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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 .

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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 .

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9. Find the angles between the lines and

Ans. Given:

Also

Let be the angle between the lines.

= = =

and

and

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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.

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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

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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

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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

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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

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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

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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,

=

=

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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

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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 .

=