Find the equation of the line passing through the points A(0, 6, – 9) and B( – 3, – 6, 3). If D is the foot of the perpendicular drawn from a point C(7, 4, – 1) on the line AB, then find the coordinates of the point D and the equation of line CD.
Given: – Line passing through the points A(0, 6, – 9) and B( – 3, – 6, 3). Point C(7, 4, – 1).
We know that
Tip: - Equation of a line joined by two points A(x1,y1,z1) and B(x2,y2,z2) is given by
Hence equation of line AB
⇒ x = – 3λ, y = – 12λ + 6, z = 12λ – 9
Now coordinates of point D
D( – 3λ, ( – 12λ + 6),( 12λ – 9))
Now as we know (TIP) ‘if two points A(x1,y1,z1) and B(x2,y2,z2) on a line, then its direction ratios are proportional to (x2 – x1,y2 – y1,z2 – z1)’
Hence
Direction ratio of CD is
= ( – 3λ – 7), ( – 12λ + 6 – 4), (12λ – 9 + 1)
= ( – 3λ – 7), ( – 12λ + 2), (12λ – 8)
and by comparing with given line equation, direction ratios of the given line are
(hint: denominator terms of line equation)
= ( – 3, – 12,12)
Since CD is perpendicular to given line, therefore by “condition of perpendicularity.”
a1a2 + b1b2 + c1c2 = 0 ; where a terms and b terms are direction ratio of lines which are perpendicular to each other.
⇒ ( – 3)( – 3λ – 7) + ( – 12)( – 12λ + 2) + 12(12λ – 8) = 0
⇒ 9λ + 21 + 144λ – 24 + 144λ – 96 = 0
⇒ 297λ – 99 = 0
Therefore coordinates of D
i.e. Foot of perpendicular
By putting the value of λ in D coordinate equation, we get
– 3λ, ( – 12λ + 6),( 12λ – 9)
Hence equation of line CD
