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Install NowExercise 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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