Find the image of the point(1,6,3) in the line x/1=(y-1)/2=(z-2)/3 Also write eq of the line joining the given point - Maths - Three Dimensional Geometry

Question

Find the image of the point(1,6,3) in the line
x/1=(y-1)/2=(z-2)/3 Also write eq of the line joining the given
point and its image and find the length of the segment joining the
given point and its image

Manbar Singh · Accepted Answer

<img alt="" src=
"https://s3mn%20mnimgs%20com/img/shared/content_ck_images/images/1(349)%20png"
style="width: 422px; height: 377px;">
Let the equation of the given line l is given byx - 01 = y - 12 = z - 23 = k sayNow, direction ratios of the given line l are:a1 = 1; b1 = 2; c1 = 3Let P1,6,3 is the point
Let N be the foot of the ⊥ drawn from the point P to the given line l
Now, general coordinates of any point on the line l are k, 2k + 1, 3k + 2
Suppose these are the coordinates of point N i
e
 Nk, 2k + 1, 3k + 2Now, direction ratios of PN are :a2 = k-1; b2 = 2k+1-6 = 2k - 5, c2 = 3k+2-3 = 3k - 1Since PN ⊥to the line l, then    a1a2 + b1b2 + c1c2 = 0⇒1k-1 + 22k-5 + 33k - 1 = 0⇒k - 1 + 4k - 10 + 9k - 3 = 0⇒14k - 14 = 0⇒14k = 14⇒k = 1Hence, coordinates of N are N1, 2×1+1,3×1+2 = N1,3,5Let Mp,q,r be the image of the point P1,6,3 in the given line lThen N must be the mid point of PM,so,  1 = p + 12 ⇒p = 1       3 = q + 62⇒q = 0       5 = r + 32 ⇒r = 7So, coordinates of point M are M1, 0, 7Hence the image of P1,6,3 in the given line l is M1, 0, 7

Find the image of the point(1,6,3) in the line x/1=(y-1)/2=(z-2)/3.Also write eq of the line joining the given point and its image and find the length of the segment joining the given point and its image.