Straight Lines
We will cover following topics
Straight Lines
General equation of the line
The general equation of a straight line in the cartesian coordinate system is given by Ax+By+C=0, where the vector (A,B) is the normal vector to the straight line.
Slope-Intercept form of the line
The slope-intercept form of the line is given by
y=mx+cwhere m is the slope and c is the y−intercept.
Equation of a line passing through two points
The equation of a straight line passing through two-points (x1,y1) and (x2,y2) is given by
y−y1y2−y1=x−x1x2−x1 or |xy1x1y11x2y21|=0
Intercept form of the Line
The intercept form of a straight line is given by
xa+yb=1where a and b are x−intercept and y−intercept respectively.
Vector equation of the Line
The vector equation of the line is given by
→r=→a+t→bwhere →a lies on the line and →b determines the direction of the line.
Point direction form of the Line
The point direction form of the line is given by
x−x1a=y−y1b=z−z1cwhere P1(x1,y1,z1) lies on the line and s=(a,b,c) is the direction vector of the line.
Distance from a Point to a Line
The distance d from a point M(x1,y1) to the line Ax+By+C=0 is given by:
d=|Ax1+By1+C|√A2+B2Condition for two lines to be Parallel
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Two straight lines y=m1x+c1 and y=m2x+c2 are parallel if k1=k2.
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Similarly, A1x+B1y+C1=0 and A2x+B2y+C2=0 are parallel if A1A2=B1B2.
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Also, if the lines are represented in point direction forms, then they are parallel if their direction vectors s1(a1,b1,c1) and s2(a2,b2,c2) are collinear, i.e., a1a2=b1b2=c1c2.
Condition for two lines to be Perpendicular
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Two straight lines y=m1x+c1 and y=m2x+c2 are perpendicular if k1k2=−1. Similarly, A1x+B1y+C1=0 and A2x+B2y+C2=0 are perpendicular if A1A2+B1B2=0.
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Also, if the lines are represented in point direction forms, then they are perpendicular if the dot product of their direction vectors is 0, i.e., s1⋅s2=0, or a1a2+b1b2+c1c2=0.
Angle between two lines
- The angle between two lines is given by:
when lines are represented in slope-intercept forms,
cosφ=A1A2+B1B2√A21+B21√A22+B22when lines are represented by their general equations, and cosφ=s1⋅s2|s1|⋅|s2|=a1a2+b1b2+c1c2√a21+b21+c21⋅√a22+b22+c22
when the lines are represented in their point direction forms.
Condition of Interesection of two lines
- Two lines x−x1a1=y−y1b1=z−z1c1 and x−x2a2=y−y2b2=z−z2c2 intersect if:
Intersection point of two lines
- The intersection point of two straight lines A1x+B1y+C1=0 and A2x+B2y+C2=0 is given by:
- The straight line given by x−x1a=y−y1b=z−z1c and the plane given by Ax+By+Cz+D=0 are:
(i) Parallel, if n⋅s=0, or Aa+Bb+Cc=0
(ii) Perpendicular, if n‖s or Aa=Bb=Cc
Shortest Distance Between Two Skew Lines
The shortest distance between two skew lines is the length of the mutual perpendicular between the two lines.
Let the two lines be given by: →r1=→a1+t⋅→b1
→r2=→a2+t⋅→b2where P=→a1 lies on l1 and Q=→a2 lies on l2.
The unit vector normal to both l1 and l2 is given by (→b1×→b2)|→b1×→b2|.
Now, the projection of PQ on the normal will be the shortest distance between l1 and l2, and is given by:
|(→a2−→a1)⋅(→b1×→b2)||→b1×→b2|PYQs
Straight Lines
1) Show that the lines x+1−3=y−32=z+21 and x1=y−7−3=z+72 intersect. Find the coordinates of the point of intersection and the equation of the plane containing them.
[10M]
2) Find the shortest distance between the lines
a1x+b1y+c1z+d1=0 a2x+b2y+c2z+d2=0and the z−axis.
[2018, 12M]
3) Find the surface generated by a line which intersects the line y=a=z,x+3z=a=y+z and parallel to the plane x+y=0.
[2016, 10M]
4) Prove that two of the straight lines represented by the equation x3+bx2y+cxy2+y3=0 will be at right angles, if b+c=−2.
[2012, 12M]
5) Find the equation of the straight line through the point (3,1,2) to intersect the straight line x+4=y+1=2(z−2) and parallel to the plane 4x+y+5z=0.
[2011, 10M]
6) A line with direction ratios 2,7,-5 is drawn to intersect the lines x1=y−12=z−24 and x−113=y−5−1=z1. Find the coordinate of the points of intersection and the length intercepted on it.
[2007, 15M]
7) Prove that the locus of a line which meets the lines y=±mx,z=±c and the circle x2+y2=a2, z=0 is c2m2(cy−mzx)2+c2(yz−cmx)2 = a2m2(z−c2)2.
[2004, 15M]
8) Find the equation of the two straight lines through the point (1,1,1) that intersect the line x−4=4(y−4)=2(z−1) at an angle of 60∘.
[2003, 12M]
Shortest Distance Between Two Skew Lines
1) Find the shortest distance between the skew the lines:
x−33=8−y1=z−31and
x+3−3=y+72=z−64[2017, 10M]
2) Find the shortest distance between the lines x−12=y−24=z−3 and y−mx=z=0. For what value of m will the two lines intersect?
[2016, 10M]
3) Show that the length of the shortest distance between the line z=xtanα,y=0 and any tangent to the ellipse x2sin2α+y2=a2,z=0 is constant.
[2006, 12M]
4) Find the equations of the lines of shortest distance between the lines: y+z=1, x=0 and x+z=1,y=0 as the intersection of two planes.
[2003, 15M]
5) Find the shortest distance between the axis of z and the lines ax+by+cz+d=0, a1x+b1y+c1z+d1=0.
[2001, 15M]