# Equation of a Line Passing through Two Given Points

## Trending Questions

**Q.**Let L be the line of intersection of planes →r⋅(^i−^j+2^k)=2 and →r⋅(2^i+^j−^k)=2. If P(α, β, γ) is the foot of perpendicular on L from the point (1, 2, 0), then the value of 35(α+β+γ) is equal to

- 143
- 101
- 134
- 119

**Q.**A line L passing through origin is perpendicular to the lines

L1:→r=(3+t)^i+(−1+2t)^j+(4+2t)^k

L2:→r=(3+2s)^i+(3+2s)^j+(2+s)^k

If the co-ordinates of the point in the first octant on L2 at the distance of √17 from the point of intersection of L and L1 are (a, b, c), then 18(a+b+c) is equal to

**Q.**Let Q be the foot of the perpendicular from the point P(7, −2, 13) on the plane containing the lines x+16=y−17=z−38 and x−13=y−25=z−37. Then (PQ)2, is equal to

**Q.**If the shortest distance between the straight lines 3(x−1)=6(y−2)=2(z−1) and 4(x−2)=2(y−λ)=(z−3), λ∈R is 1√38, then the integral value of λ is equal to:

- 2
- 5
- 3
- −1

**Q.**The line passing through the points (5, 1, a) and (3, b, 1) crosses the yz-plane at the point (0, 172, −132), Then

- a = 2, b = 8
- a = 4, b = 6
- a = 6, b = 4
- a = 8, b = 2

**Q.**A line L passing through the point P(1, 4, 3) is perpendicular to both the lines

x−12=y+31=z−24 and x+23=y−42=z+1−2.

If the position vector of the point Q on L is (a1, a2, a3) such that PQ2=357, then (a1+a2+a3) can be

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- 16
- 2
- 1

**Q.**If the acute angle between the line →r=^i+2^j+λ(4^i−3^k) and xy−plane is α and the acute angle between the planes x+2y=0 and 2x+y=0 is β, then (cos2α+sin2β) equals

**Q.**If a point R(4, y, z) lies on the line segment joining the points P(2, −3, 4) and Q(8, 0, 10), then the distance of R from the origin is :

- √53
- 2√14
- 2√21
- 6

**Q.**The plane 4x + 7y + 4z + 81 = 0 is rotated through a right angle about its line of intersection with the plane 5x + 3y + 10z = 25. The equation of the plane in its new position is x – 4y +6z = k, where k is

- 73
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- -89
- 106

**Q.**

Find the distance of the point (- 1, - 5, - 10) from the point of intersection of the line r=2^i−^j+2^k+λ(3^i+4^j+2^k) and the plane r.(^i−^j+^k)=5

**Q.**

Find the Cartesian equation of the line which passes through the point (- 2, 4, -5) and parallel to the line given by x+33=y−45=z+86

**Q.**Find the coordinates of the point where the line through (5, 1, 6) and (3, 4, 1) crosses the YZ -plane.

**Q.**Find the distance of the point (-2, 3, -4) from the line

x+2/3 = 2y+3/4=3z+4/5 measure parallel to the plane

4x+12y-3z+1=0.

**Q.**The distance between the line x=2+t, y=1+t, z=−12−t2 where t∈R and the plane →r⋅(^i+2^j+6^k)=10 is

- 16
- 1√41
- 17
- 9√41

**Q.**Find the equation of the line through the points (3, 4, -7) and (1, -1, 6)

**Q.**Find the distance of the point P(3, 4, 4) from the point, where the line joining the points A(3, −4, −5) and B(2, −3, 1) intersects the plane 2x + y + z = 7. [CBSE 2015]

**Q.**

Find the coordinates of
the point where the line through (3, Ââˆ’4, âˆ’5) and (2,
âˆ’ 3, 1) crosses the plane 2*x* + *y *+ *z* = 7).

**Q.**

If 0 is the origin and A is (a, b, c), then find the direction cosines of the line OA and the equation of plane through A at right angle OA.

**Q.**

Find the orthocecentre of the triangle with the vertices (-5, -7), (13, 2), (-5, 6)

**Q.**

Determine the direction cosines of the normal to plane and the distance from the origin:

5y + 8 = 0

**Q.**

P (*a,
b*) is the mid-point of a line segment between axes. Show that
equation of the line is

**Q.**The distance of the point P(3, 4, 4) from the point of intersection of the line joining the points Q(3, –4, –5) and R(2, –3, 1) and the plane

2x+y+z=7, is equal to

**Q.**A line L passing through the point P(1, 4, 3) is perpendicular to both the lines x−12=y+31=z−24 and x+23=y−42=z+1−2 . If the position vector of the point Q on L is (a1, a2, a3) such that PQ2=357, then (a1+a2+a3) can be

- 15
- 1
- 2
- 16

**Q.**

Find the equation of the plane which contains line of intersection of the planes r.(^i+2^j+3^k)−4=0, r.(2^i+^j−^k)+5=0 and which is perpendicular to the plane r.(5^i+3^j−6^k)+8=0 .

**Q.**Find the vector and the Cartesian equations of the line that passes through the points (3, −2, −5), (3, −2, 6).

**Q.**Using differentials, find the approximate value of each of the following up to 3 places of decimal.

(i) √25.3

(ii) √49.5

(iii) √0.6

(iv) (0.009)13

(v) (0.999)110

(vi) (15)14

(vii) (26)13

(viii) (255)14

(ix) (82)14

(x) (401)12

(xi) (0.0037)12

**Q.**The coordinates of the foot of the perpendicular from the points (1, −2, 1) on the plane containing the lines, x+16=y−17=z−38 and x−13=y−25=z−37 is:

- (2, −4, 2)
- (0, 0, 0)
- (−1, 2, −1)
- (1, 1, 1)

**Q.**Let P be the foot of the perpendicular dropped from the origin O on the line of intersection of the planes x−2y+3z=5 and 2x+3y+z+4=0, then

- Equation of the plane perpendicular to the given line and passing through the point (1, 1, 1) is 11x−5y−7z+1=0
- Equation of acture angle bisector of the two planes is x+5y−2z+9=0.
- ¯¯¯¯¯¯¯¯OP, ^i−2^j+3^k and 2^i+3^j+^k must be coplanar
- The point (1, 2, 3) lies in acute region of the two planes

**Q.**If the distance of the point (1, −2, 3) from the plane x+2y−3z+10=0 measured parallel to the line, x−13=2−ym=z+31 is √72, then the value of |m| is equal to

**Q.**Find the vector and the Cartesian equations of the lines that pass through the origin and (5, −2, 3).