Exercise 9.1 [BCB pg 244]

Solution 👉 Click Here

1. Find the lengths of perpendiculars drawn from
   1. (0,0) to the line 3x+y+1=0
   2. (-3,0) to the line 3x+4y+7=0
   3. (2,3) to the line 8x+15y+24=0
2. If p is the length of the perpendicular dropped from the origin on the line \(\frac{x}{a}+\frac{y}{b}=1\) prove that \(\frac{1}{a^2}+\frac{1}{b^2}=\frac{1}{p^2}\)
3. Find the distance between the parallel lines
   1. 3x+5y=11 and 3x+5y=-23
   2. 2x-5y=6 and 6x-15y+11=0
4. Find the equation of bidectors of the angles between the lines
   1. 3x-4y+2=0 and 5x+12y+5=0
   2. 3x-4y+6=0 and 5x+12y+10=0
   3. x-2y=0 and 2y-11x=6
5. 
   
   1. Find the equation of the bisectors of the angles between the lines containing the origin in each of the following cases
      1. 4x-3y+1=0 and 12x-5y+7=0
      2. 7x-y+11=0 and x+y-15=0
      Prove that the bisectors of angles are at right angles to each other.
   2. Find the equation of the bisectors of acute angles between each of the following pair of lines
      1. y=x and y=7x+4
      2. x+2y=5 and 4x+2y+9=0
      Prove that the bisectors of angles are at right angles to each other.
6. 
   
   1. Are the points (1,2) and (-5,6) on the same side or on opposite side of the line 3x+5y-8=0?
   2. On what side of the line 5x-4y+6=0 do the points (0,0) and (-1,3) lie?
   3. Show that two of the three points (0,0),(2,3) and (3,4) lie on one side and the remaining on the other side of the line x-3y+3=0
7. 
   
   1. The length of the perpendicular drawn from the point (a,3) on the line 3x+4y+5=0 is 4. Find the value of a.
   2. What are the points on the axis of X whose perpendicular distance from the striaght line \(\frac{x}{a}+\frac{y}{b}=1\) is a ?
8. 
   
   1. Determine the equation and length of the altitude drawn from the vertex A to the opposite side of the triangle A(1,0), B(1,3), C(4,-2).
   2. Find the equation of two straight lines each of which is parallel to and at a distance of \(\sqrt{5}\) from the line x+2y-7=0
   3. Find the equations of the two straight lines drawn through the point (0,a) on which the perpendicular drawn from the point (2a,2a) are each of length a
   4. Find the equation of line which is at right angles to 3x+4y=12 such that its perpendicular distance from the origin is equal to the length of the perpendicular from (3,2) on the given line.
   5. The equation of the diagonal of a parallelogram is 3y=5x+k. The two opposite vertices of a parallelogram are the points (1,-2) and (-2,1). Find the value of k.
9. If p and p' be the length of the perpendiculars from the origin upon the straight line whose equations are \(\sec \theta + y \csc \theta = a \) and \(x \cos \theta - y \sin \theta = a \cos 2 \theta \), prove that \(4p^2 + p'^2 = a^2\)
10. Show that the product of the perpendiculars drawn from the two points \( ( \pm \sqrt{a^2-b^2},0) \) upon the line \(\frac{x}{a}\cos \theta +\frac{y}{b} \sin \theta =1\) is \(b^2\)
11. The origin is a corner of a square and two of its sides are y + 2x = 0 and y + 2x = 3. Find the equation of the other two sides.
12. 
    
    1. A triangle is fomed by lines x+ y= 6, 7x - y + 10 = 0 and 3x + 4y + 9 = 0. Find the equation of the internal bisector of the angles between the first two sides.
    2. Find the equations of the internal bisectors of the angles of the triangle whose sides are 4x - 3y + 2 = 0, 3x - 4y + 12 = 0 and 3x + 4y - 12 = 0. Also find the incentre of the triangle.