{"id":4616,"date":"2016-05-17T09:49:00","date_gmt":"2016-05-17T04:19:00","guid":{"rendered":"http:\/\/mycbseguide.com\/blog\/ncert-solutions-class-11-maths-miscellaneous-8\/"},"modified":"2016-05-17T11:26:52","modified_gmt":"2016-05-17T05:56:52","slug":"ncert-solutions-class-11-maths-miscellaneous-8","status":"publish","type":"post","link":"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-11-maths-miscellaneous-8\/","title":{"rendered":"NCERT Solutions class-11 Maths Miscellaneous"},"content":{"rendered":"

Miscellaneous Exercise<\/strong><\/p>\n

1. Find the values of K for which the line <\/strong> is:<\/strong><\/p>\n

(a) parallel to the axis <\/strong><\/p>\n

(b) parallel to the axis <\/strong><\/p>\n

(c) passing through the origin.<\/strong><\/p>\n

Ans. <\/strong>Given: Equation of line <\/p>\n

<\/p>\n

(a) <\/strong>If the line parallel to axis, then <\/p>\n

<\/p>\n

<\/p>\n

<\/p>\n

(b) <\/strong>If the line parallel to axis, then <\/p>\n

<\/p>\n

<\/p>\n

<\/p>\n

<\/p>\n

(c)<\/strong> If the line passes through origin then<\/p>\n

<\/p>\n

<\/p>\n

<\/p>\n

K = 1 or K = 6<\/p>\n

\n

2. Find the values of <\/strong> and <\/strong> if the equation <\/strong> is the normal form of the line <\/strong><\/p>\n

Ans. <\/strong>Given: <\/p>\n

<\/p>\n

<\/p>\n

Dividing both sides by , we have <\/p>\n

Putting and ,<\/p>\n

<\/p>\n

<\/p>\n

Equation of line in normal form <\/p>\n

Comparing with and <\/p>\n

\n

3. Find the equations of the lines which cut-off intercepts on the axes whose sum and product are 1 and <\/strong> respectively.<\/strong><\/p>\n

Ans. <\/strong>Let equation of line be and it is given that and <\/p>\n

<\/p>\n

<\/p>\n

<\/p>\n

<\/p>\n

Now, solving and , <\/p>\n

Now, solving and <\/p>\n

Therefore, the required line <\/p>\n

<\/p>\n

<\/p>\n

And <\/p>\n

<\/p>\n

<\/p>\n

\n

4. What are the points on the <\/strong>axis whose distance from the line <\/strong> is 4 units?<\/strong><\/p>\n

Ans. <\/strong>Given: Equation of line and let point on axis be <\/p>\n

<\/p>\n

<\/p>\n

Perpendicular distance from point to this line = = <\/p>\n

According to question <\/p>\n

<\/p>\n

Taking <\/p>\n

<\/p>\n

<\/p>\n

Taking <\/p>\n

<\/p>\n

<\/p>\n

Therefore, required points are and <\/p>\n

\n

5. Find the perpendicular distance from the origin of the line joining the points <\/strong> and <\/strong><\/p>\n

Ans. <\/strong>Equation of the line joining points and <\/p>\n

<\/p>\n

<\/p>\n

<\/p>\n

<\/p>\n

Now, perpendicular distance from (0, 0) to this line,<\/p>\n

= <\/p>\n

= <\/p>\n

= <\/p>\n

= <\/p>\n

= <\/p>\n

= <\/p>\n

\n

6. Find the equation of the line parallel to <\/strong>axis and drawn through the point of intersection of the lines <\/strong> and <\/strong><\/p>\n

Ans. <\/strong>Solving and , we have and <\/p>\n

Point of intersection of lines are and <\/p>\n

But the required line is parallel to axis, therefore the equation of required line is .<\/p>\n

\n

7. Find the equation of a line drawn perpendicular to the line <\/strong> through the point, where it meets the <\/strong>axis.<\/strong><\/p>\n

Ans. <\/strong>Given: Equation of line <\/p>\n

<\/p>\n

Slope of given line = <\/p>\n

Since the required line is perpendicular to given line.<\/p>\n

Slope of required line = <\/p>\n

The given line meets the axis at (0, 6).<\/p>\n

Equation of required line is <\/p>\n

<\/p>\n

<\/p>\n

\n

8. Find the area of the triangle formed by the lines <\/strong> and <\/strong><\/p>\n

Ans. <\/strong>Given: Equations of lines are \u2026..(i)<\/p>\n

\u2026..(ii)<\/p>\n

<\/p>\n

And \u2026..(iii)<\/p>\n

On solving eq. (i) and (ii), we get the coordinates of point C (0, 0)<\/p>\n

On solving eq. (ii) and (iii), we get the coordinates of point A<\/p>\n

On solving eq. (i) and (iii), we get the coordinates of point B<\/p>\n

Area of = <\/p>\n

= <\/p>\n

= <\/p>\n

= sq. units<\/p>\n

\n

9. Find the value of <\/strong> so that three lines <\/strong> and <\/strong> may intersect at one point.<\/strong><\/p>\n

Ans. <\/strong>Given: Equations of lines are and <\/p>\n

Three lines are concurrent if <\/p>\n

<\/p>\n

<\/p>\n

<\/p>\n

<\/p>\n

<\/p>\n

\n

10. If three lines whose equations are <\/strong> and <\/strong> are concurrent., then show that <\/strong><\/p>\n

Ans. <\/strong>Given: Equations of lines are and <\/p>\n

Three lines are concurrent if <\/p>\n

<\/p>\n

<\/p>\n

\n

11. Find the equations of the lines through the point (3, 2) which make an angle of <\/strong> with the line <\/strong><\/p>\n

Ans. <\/strong>Let be the slope of required line which passes through point (3, 2),<\/p>\n

Then the equation of required line is \u2026\u2026\u2026.(i)<\/p>\n

The equation of given line <\/p>\n

\u2026\u2026\u2026.(ii)<\/p>\n

Slope of given line = <\/p>\n

According to question, <\/p>\n

<\/p>\n

<\/p>\n

Taking <\/p>\n

<\/p>\n

<\/p>\n

Then the equation of required line is <\/p>\n

<\/p>\n

<\/p>\n

Taking <\/p>\n

<\/p>\n

<\/p>\n

Then the equation of required line is <\/p>\n

<\/p>\n

<\/p>\n

\n

12. Find the equation of the line passing through the point of intersection of the lines <\/strong> and <\/strong> that has equal intercepts on the axis.<\/strong><\/p>\n

Ans. <\/strong>Given: Equations of lines and <\/p>\n

Equation of any line through intersection of these lines <\/p>\n

<\/p>\n

<\/p>\n

<\/p>\n

But, according to question <\/p>\n

<\/p>\n

or <\/p>\n

or <\/p>\n

or <\/p>\n

Equation of any line through intersection of these lines <\/p>\n

<\/p>\n

<\/p>\n

<\/p>\n

Also Equation of any line through intersection of these lines <\/p>\n

<\/p>\n

<\/p>\n

<\/p>\n

\n

13. Show that the equation of the line passing through the origin and making an angle <\/strong> with the line <\/strong> is <\/strong><\/p>\n

Ans. <\/strong>Let be the slope of required line which passes through (0, 0).<\/p>\n

Equation of line is <\/p>\n

\u2026\u2026\u2026(i)<\/p>\n

Now, is the angle between and <\/p>\n

<\/p>\n

<\/p>\n

or <\/p>\n

or <\/p>\n

or <\/p>\n

or <\/p>\n

<\/p>\n

Putting the value of in eq. (i), we get<\/p>\n

<\/p>\n

<\/p>\n

<\/p>\n

\n

14. In what ratio, the line joining <\/strong> and (5, 7) is divided by the line <\/strong> ?<\/strong><\/p>\n

Ans. <\/strong>Given: Equation of line <\/p>\n

Let the given line divide the line joining A and B (5, 7) in the ratio at a point C.<\/p>\n

<\/p>\n

Coordinates of C are <\/p>\n

Since the point C lies on the given line.<\/p>\n

<\/p>\n

<\/p>\n

<\/p>\n

<\/p>\n

Therefore, the required ratio is 1: 2.<\/p>\n

\n

15. Find the distance of the line <\/strong> from the point (1, 2) along the line <\/strong><\/p>\n

Ans. <\/strong>Here point of intersection of given line and are and <\/p>\n

Distance between the points (1, 2) and <\/p>\n

= <\/p>\n

= <\/p>\n

= units<\/p>\n

\n

16. Find the direction in which a straight line must be drawn through the point <\/strong> so that its point of intersection with the line <\/strong> may be at a distance of 3 units from this point.<\/strong><\/p>\n

Ans. <\/strong>Let the required line makes an angle with the positive direction of axis.<\/p>\n

Equation of the line is <\/p>\n

<\/p>\n

(given)<\/p>\n

and <\/p>\n

and <\/p>\n

Since this point lies on <\/p>\n

<\/p>\n

<\/p>\n

<\/p>\n

Squaring both side, we have <\/p>\n

<\/p>\n

<\/p>\n

<\/p>\n

<\/p>\n

<\/p>\n

Therefore, the required line is parallel to axis or parallel to axis.<\/p>\n

\n

17. The hypotenuse of a right angled triangle has its ends at the points (1, 3) and <\/strong> Find the equation of the legs (perpendicular sides) of the triangle.<\/strong><\/p>\n

Ans. <\/strong>Let ABC be a right angled triangle with diagonal AC.<\/p>\n

<\/p>\n

Slope of AC = <\/p>\n

coordinate of A = coordinate of C = 1<\/p>\n

AB = AC<\/p>\n

A = C = <\/p>\n

Let be the slope of lines with diagonal AC.<\/p>\n

<\/p>\n

<\/p>\n

<\/p>\n

Taking <\/p>\n

<\/p>\n

<\/p>\n

<\/p>\n

Taking <\/p>\n

<\/p>\n

<\/p>\n

<\/p>\n

Equation of line AB and <\/p>\n

and <\/p>\n

and <\/p>\n

Equation of line BC and <\/p>\n

and <\/p>\n

and <\/p>\n

\n

18. Find the image of the point (3, 8) with respect to the line <\/strong> assuming the line to be a plane mirror.<\/strong><\/p>\n

Ans. <\/strong>Let the image of the point A (3, 8) in the line mirror DE be C Then AC is perpendicular bisector of DE.<\/p>\n

<\/p>\n

The coordinates of point B are <\/p>\n

Since point B lies on the line <\/p>\n

<\/p>\n

<\/p>\n

\u2026..(i)<\/p>\n

Since AC is perpendicular on DE.<\/p>\n

Slope of AC x Slope of DE = <\/p>\n

<\/p>\n

<\/p>\n

\u2026..(ii)<\/p>\n

Solving eq. (i) and (ii), we get and <\/p>\n

Therefore, the image of point (3, 8) is <\/p>\n

\n

19. If the lines <\/strong> and <\/strong> are equally inclined to the line <\/strong> find the value of <\/strong><\/p>\n

Ans. <\/strong>Let be the angle which the line makes with the line and <\/p>\n

and <\/p>\n

<\/p>\n

<\/p>\n

<\/p>\n

or <\/p>\n

<\/p>\n

\n

20. If sum of the perpendicular distances of a variable point P<\/strong> from the lines <\/strong> and <\/strong> is always 10. Show that P must move on a line.<\/strong><\/p>\n

Ans. <\/strong>Given: Equations of lines are \u2026..(i)<\/p>\n

and \u2026..(ii)<\/p>\n

Perpendicular distance of point P from line (i)<\/p>\n

= <\/p>\n

Perpendicular distance of point P from line (ii)<\/p>\n

= <\/p>\n

According to question, <\/p>\n

<\/p>\n

When and <\/p>\n

Then <\/p>\n

, which represent a line.<\/p>\n

Similarly, and and <\/p>\n

and <\/p>\n

In all the above cases, it represents a line therefore, P must move on a line.<\/p>\n

\n

21. Find equation of the line which is equidistant from parallel lines <\/strong> and <\/strong><\/p>\n

Ans. <\/strong>The equations of parallel lines are and <\/p>\n

Let A be any point which is equidistant from the parallel lines.<\/p>\n

<\/p>\n

<\/p>\n

Taking <\/p>\n

<\/p>\n

<\/p>\n

Therefore, the required line is <\/p>\n

\n

22. A ray of light passing through the point A and the reflected ray passes through the point (1, 2) reflects on the <\/strong>axis at point A and the reflected ray passes through the point (5, 3). Find the coordinates of A.<\/strong><\/p>\n

Ans. <\/strong>Let BA be the incident ray and AC be the reflected ray.<\/p>\n

<\/p>\n

Now, for line AC <\/p>\n

\u2026..(i)<\/p>\n

For line BA <\/p>\n

\u2026..(ii)<\/p>\n

From eq. (i) and (ii), <\/p>\n

<\/p>\n

<\/p>\n

<\/p>\n

Therefore, coordinates of point A are <\/p>\n

\n

23. Prove that the product of the lengths of the perpendiculars drawn from the points <\/strong> and <\/strong> to the line <\/strong> is <\/strong><\/p>\n

Ans. <\/strong>Let and be the length of perpendiculars from and to the line <\/p>\n

<\/p>\n

= <\/p>\n

And <\/p>\n

= <\/p>\n

Now <\/p>\n

= <\/p>\n

= <\/p>\n

= <\/p>\n

= <\/p>\n

= = <\/p>\n

\n

24. A person standing at the junction (crossing) of two straight paths represented by the equations <\/strong> and <\/strong> wants to reach the path whose equation is <\/strong> in the least time. Find equation of the path that he should follow.<\/strong><\/p>\n

Ans. <\/strong>The point of intersection of lines and if given by Since the shortest path through point A is perpendicular line AB.<\/p>\n

<\/p>\n

Slope of line is <\/p>\n

Slope of required line is <\/p>\n

Therefore, the equation of required line is<\/p>\n

<\/p>\n

<\/p>\n

<\/p>\n","protected":false},"excerpt":{"rendered":"

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 … Read more<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[47,281],"tags":[],"class_list":["post-4616","post","type-post","status-publish","format-standard","hentry","category-cbse-class-11","category-ncert-solutions"],"yoast_head":"\nNCERT Solutions class-11 Maths Miscellaneous | myCBSEguide<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-11-maths-miscellaneous-8\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"NCERT Solutions class-11 Maths Miscellaneous | myCBSEguide\" \/>\n<meta property=\"og:description\" content=\"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 ... 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