Question

Find the value of t, the points ( 2,-1, 3),(3,-5 , t) and (-1, 11, 9) are collinear.​

Answer 1

Answer:

Given points, (2,−1,3),(3,−5,1) and (−1,11,9)

Three points A,B,C are collinear if direction ratios of AB and BC are proportional

A≡(2,−1,3);B≡(3,−5,1);C≡(−1,11,9)

AB→ Direction ratios ≡(1,−4,−2)

BC→ Direction ratios ≡(−4,16,8)

∴a

1

=1,b

1

=−4,c

1

=−2

a

2

=−4,b

2

=16,c

2

=8

∴

a

2

a

1

=

b

2

b

1

=

c

2 c 1 = 4−1

Therefore A,B,C are collinear

Answer 2

The second derivative of an implicit function can be found using sequential differentiation of the initial equation F(x,y)=0. At the first step, we get the first derivative in the form y′=f1(x,y). On the next step, we find the second derivative, which can be expressed in terms of the variables x and y as y′′=f2(x,y).