A common tangent to the conics x2=6y and 2x24y2=9, is
Answers
Answer:
Step-by-step explanation:
Find all points where the equations given INTERSECT and have the same SLOPE
... which is the same as saying where the equations share the same tangent line.
Note: dy/dx = ý = slope of an equation.
x² = 6y 2x² – 4y² = 9 ——>> y = ± √ [(2x² – 9) ⁄ 4 ]
2x = 6ý 4x – 8yý = 0
ý = x ⁄ 3 ý = x ⁄ (2y)
Setting: ý = ý :
x ⁄ 3 = x ⁄ (2y)
It looks like x=0 is one of the solutions, since the two initial equations appear
to have the same slope at: x=0
First I'll check if the two initial equation even intersect at: x=0 :
Using the first initial equation: x² = 6y , when: x=0, then: y=0
Using the second equation: 2x² – 4y² = 9 , when: x=0, then: y=imaginary number
... therefore the two equations DO NOT INTERSECT at: x=0
Getting back to the derivative equations:
1st equation derivative = 2nd equation derivative
x ⁄ 3 = x ⁄ (2y) ... recall: (x=0) is NOT a solution ... divide by "x"
1 ⁄ 3 = 1 ⁄ (2y) ... where "y" refers to the 2nd initial equation (see above)
y = 3 ⁄ 2 ... now substitute for "y"
± √ [(2x² – 9) ⁄ 4 ] = 3 ⁄ 2
± √ [(2x² – 9) ⁄ 4 ] = 3 ⁄ 2
(2x² – 9) ⁄ 4 = 9 ⁄ 4
2x² – 9 = 9
2x² = 18
x = ± 3 ... locations where the slopes of each
... equation are equivalent.
Now I'll check to see if the two initial equations intersect at: x = ± 3
... at: x = 3 , for equation: x² = 6y ——>> y = 3/2
... and at: x = 3 , for equation: y = ± √ [(2x² – 9) ⁄ 4 ] ——>> y = ± 3/2
... so they intersect at: (x, y) = (3, 3/2)
... at: x = - 3 , for equation: x² = 6y ——>> y = 3/2
... and at: x = - 3 , for equation: y = ± √ [(2x² – 9) ⁄ 4 ] ——>> y = ± 3/2
... so they intersect at: (x, y) = (- 3, 3/2)
Recall the two derivative equations are: (from above):
ý = x ⁄ 3 ý = x ⁄ (2y)
ý{- 3} = -3 ⁄ 3 ý{- 3} = -3 ⁄ (2 • (3/2))
ý{- 3} = -1 ý{- 3} = -1
ý = x ⁄ 3 ý = x ⁄ (2y)
ý{3} = 3 ⁄ 3 ý{3} = 3 ⁄ (2 • (3/2))
ý{3} = 1 ý{3} = 1
Therefore the two initial equations share a common tangent line:
at: (- 3, 3/2) , with the same slope = ý{-3} = -1
and (3, 3/2) , with the same slope = ý{3} = 1
Answer:TAKE IT AS MUCH AS U WANT BUT FAST AS U COULD GOOD BYE
Step-by-step explanation: