Find the vector parametric equations of the two straight lines tangent to the gi

Question

Find the vector parametric equations of the two straight lines tangent to the given space curve and which pass through the point P(-8, -15,-34) (not on the curve). r(t)=(a2+1)i+ (11t-4)I + (12t2+2)k Let s1 be a real number parameter. Give the parameterized vector equation of the first tangent line r(s1)=(-8,-15,-34) +S1(6 (Type integers or simplified fractions.) Let s2 be a real number parameter. Give the parameterized vector equation of the second tangent line. r(s,) = +s-18 Type integers or simplified fractions.)

Explanation / Answer

Lets derive r :
r' = <6t , 11 , 24t>

Clearly equating 6 = 6t, we get t = 1

So, we have
r' = <6t , 11 , 24t>
when we put t = 1, we get :
r'(1) = <6 , 11 , 24>

So, first tangent line is :
<-8,-15,-34> + s1<6,11,24> ---> ANS

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r' = <6t , 11 , 24t>
Now, equating : 6t = -18
So, t = -3

So, r'(-3) = <-18 , 11 , -72>

And for the point,
11t - 4 = -37
11t = -33
t = -3 which of course matches

So with r = <3t^2 + 1 , 11t - 4, 12t^2 + 2>
we get point r(-3) = (28 , -37 , 110)

And thus, ans :
(28 , -37 , 110) + s2<-18 , 11 , -72> ----> ANS

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