Find the equation of the perpendicular drawn from the point P(2, 4, –1) to the line
Also, write down the coordinates of the foot of the perpendicular from P.
Given: - Point P(2, 4, – 1) and equation of line
Let, PQ be the perpendicular drawn from P to given line whose endpoint/ foot is Q point.
Thus to find Distance PQ we have to first find coordinates of Q
⇒ x = λ – 5, y = 4λ – 3, z = – 9λ + 6
Therefore, coordinates of Q(λ – 5,4λ – 3, – 9λ + 6)
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 PQ is
= (λ – 5 – 2), (4λ – 3 – 4), ( – 9λ + 6 + 1)
= (λ – 7), (4λ – 7), ( – 9λ + 7)
and by comparing with given line equation, direction ratios of the given line are
(hint: denominator terms of line equation)
= (1,4, – 9)
Since PQ 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.
⇒ 1(λ – 7) + (4)(4λ – 7) – 9( – 9λ + 7) = 0
⇒ λ – 7 + 16λ – 28 + 81λ – 63 = 0
⇒ 98λ – 98 = 0
⇒ λ = 1
Therefore coordinates of Q
i.e. Foot of perpendicular
By putting the value of λ in Q coordinate equation, we get
Now,
So, Equation of perpendicular PQ is
Tip: - Equation of a line joined by two points A(x1,y1,z1) and B(x2,y2,z2) is given by