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G Gautam harsolia
we know that intercept form of line is we know that In this problem On squaring both the sides  we will get Hence proved

G Gautam harsolia
Let suppose foot of perpendicular is  We can say that line passing through point   is perpendicular to line passing through point  Now, Slope of line passing through point  is ,  And Slope of line  passing through point  is ,  lines are perpendicular Therefore, Now,  equation of line passing through point  (2 ,3)  and slope with 1                      -(i) Now,...

G Gautam harsolia
Given equations of lines are      and   We can rewrite the equation  as Now, we know that  In equation   Similarly, in the equation    Now,                                                                                                                                                                                                     Hence proved

G Gautam harsolia
We can say that line passing through point   is perpendicular to line  Now, The slope of the line  passing through the point  is ,  lines are perpendicular Therefore,                - (i) Now, the point  also lies on the line  Therefore, Therefore, the value of m and C is    respectively

G Gautam harsolia
Let suppose the foot of perpendicular is  We can say that line passing through the point   is perpendicular to the line  Now, The slope of the line  is ,  And The slope of the line  passing through the point is,  lines are perpendicular Therefore, Now, the point  also lies on the line  Therefore, On solving equation (i) and (ii) we will get Therefore,

G Gautam harsolia
Right bisector means perpendicular line which divides the line segment into two equal parts Now, lines are perpendicular which means their slopes are negative times inverse of each other Slope of line passing through points    and     is Therefore, Slope of bisector line is Now, let (h , k) be the point of intersection of two lines  It is given that point (h,k) divides the line segment...

G Gautam harsolia
Let the slope of two lines are    respectively It is given the lines intersects each other at an angle of    and slope of the line is 2 Now, Now, the equation of line passing through point (2 ,3) and with slope    is                          -(i) Similarly, Now , equation of line passing through point (2 ,3) and with slope    is                               -(ii) Therefore,...

G Gautam harsolia
It is given that line is parallel to the line   Therefore, their slopes are equal The slope of line   ,  Let the slope of other line be m Then, Now, the equation of the line passing through the point   and with slope   is Hence proved

G Gautam harsolia
Line passing through points ( h ,3) and (4 ,1) Therefore,Slope of the line is   This line intersects the line   at right angle Therefore, the Slope of both the lines are negative times inverse of each other  Slope of line  , Now, Therefore, the value of h is

G Gautam harsolia
Given equation of lines are     and    Slope of line  is,  And  Slope of line   is ,  Now, if   is the angle between the lines Then, Therefore, the angle between the lines is

G Gautam harsolia
It is given that line is  perpendicular to the line  we can rewrite it as Slope of line    ( m' ) =  Now,  The slope of the line is       Now, the equation of the line with  intercept   i.e. (3, 0) and  with slope -7 is

G Gautam harsolia
It is given that line is parallel to line   which implies that the slopes of both the lines are equal we can rewrite it as The slope of line   =   Now, the equation of the line passing through the point  and with slope  is Therefore, the equation of the line is

G Gautam harsolia
Given equations of lines are  and  it is given that these lines are parallel Therefore, Now, Therefore, the distance between two lines is

G Gautam harsolia
Given equations of lines are    and   it is given that these lines are parallel Therefore, Now, Therefore, the distance between two lines is

G Gautam harsolia
Given equation of line is we can rewrite it as Now, we know that In this problem A = 4 , B = 3 C = -12 and d = 4 point is on x-axis therefore   = (x ,0) Now, Now if x > 3 Then,   Therefore, point is (8,0)            and if x < 3 Then, Therefore, point is (-2,0) Therefore, the points on the x-axis, whose distances from the line    are    units are  (8 , 0) and (-2 , 0)

G Gautam harsolia
Given the equation of the line is we can rewrite it as Now, we know that         where A and B are the coefficients of x and y and  C is some constant  and   is point from which we need to find the distance  In this problem A = 12 , B = -5 , c = 82 and  = (-1 , 1) Therefore, Therefore, the distance of the point    from the line    is 5 units

G Gautam harsolia
Given equation is Coefficient of x is 1 and y is -1 Therefore,  Now, Divide both the sides by  we wiil get we can rewrite it as Now, we know that normal form of line is Where  is the angle between perpendicular and the positive x-axis and p is the perpendicular distance  from the origin On compairing equation (i) and (ii) we wiil get Therefore,  the angle between perpendicular...

G Gautam harsolia
Given equation is we can rewrite it as Coefficient of x is 0 and y is 1 Therefore,  Now, Divide both the sides by 1 we will get we can rewrite it as Now, we know that normal form of line is Where  is the angle between perpendicular and the positive x-axis and p is the perpendicular distance  from the origin On comparing equation (i) and (ii) we wiil get Therefore,  the...