Find the perpendicular distance of the point (1, 0, 0) from the line Also, find the coordinates of the foot of the perpendicular and the equation of the perpendicular.
Given: - Point P(1, 0, 0) 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 = 2λ + 1, y = – 3λ – 1, z = 8λ – 10
Therefore, coordinates of Q(2λ + 1, – 3λ – 1,8λ – 10)
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
= (2λ + 1 – 1), ( – 3λ – 1 – 0), (8λ – 10 – 0)
= (2λ), ( – 3λ – 1), (8λ – 10)
and by comparing with given line equation, direction ratios of the given line are
(hint: denominator terms of line equation)
= (2, – 3,8)
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.
⇒ 2(2λ) + ( – 3)( – 3λ – 1) + 8(8λ – 10) = 0
⇒ 4λ + 9λ + 3 + 64λ – 80 = 0
⇒ 77λ – 77 = 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,
Distance between PQ
Tip: – Distance between two points A(x1,y1,z1) and B(x2,y2,z2) is given by
= 2√6 unit