CONCEPT: - The direction ratios of the line joining the points
(x1,y1,z1) and
(x2,y2,z2) is given by:
a=x2−x1,b=y2−y1 and
c=z2−z1 - If
a,b,c are the direction ration ratios of a line passing throughthe point
(x1,y1,z1), then the equation of line is given by:
‌=‌=‌ CALCULATION: Given: The line joining the points
(k,1,3) and
(1,−2,k+1) also passes through the point
(15,2,−4) As we knowthat, the direction ratios of the line joining the points
(x1,y1,z1) and
(x2,y2,z2) is given by:
a=x2−x1,b=y2−y1 and
c=z2−z1 So, the direction ratios of the line joining the points
(k,1,3) and
(1,−2,k+1) is:
a=1−k,b=−3 and
c=k−2 As we know that, if
a,b,c are the direction ration ratios of a line passing through the point
(x1,y1,z1),
then the equation of line is given by:
‌=‌=‌ So, the equation of the line with direction ratios
a,b,c and passing through the the point
(k,1,3) is given by:
‌=‌=‌ ∵ It is giventhat, the line represented by
‌=‌=‌‌‌ also passes through the point
(15,2,−4) So, substitute
x=15,y=2 and
z=−4 in the equation
‌=‌=‌ ⇒‌=‌=‌ ⇒‌=‌ ⇒−45+3k=1−k ⇒k=23∕2 ⇒‌=‌ ⇒k−2=21 ⇒k=23 So, there are two possible values of
k as shown above.
Hence, correct option is 3