Explanation The plane intersects only one nap of the double right circular cone as long as the angle between the plane and the vertical axis is greater than or equal to the vertex angle. However, if the plane intersects the vertical axis at an angle smaller than the vertex angle, the plane intersects both the naps of the cone to form an open curve called a hyperbola which has two disjoint curves. Given: P F 1 − P F 2 = ± 2 a with foci F 1 = ( c , 0 ) and F 2 = ( − c , 0 ) and c = a 2 + b 2 . Formula Used: Assume P ( x , y ) is a point on the hyperbola. So, P F 1 = ( x − c ) 2 + y 2 and P F 2 = ( x + c ) 2 + y 2 . Calculation: It is given that, P F 1 − P F 2 = ± 2 a then, P F 1 − P F 2 = ± 2 a ( x − c ) 2 + y 2 − ( x + c ) 2 + y 2 = ± 2 a ( x − c ) 2 + y 2 = ± 2 a + ( x + c ) 2 + y 2 x 2 − 2 c x + c 2 + y 2 = 4 a 2 + x 2 + 2 c x + c 2 + y 2 ± 4 a ( x + c ) 2 + y 2 − 4 c x − 4 a 2 = ± 4 a ( x + c ) 2 + y 2 − 4 ( c x + a 2 ) = ± 4 a ( x + c ) 2 + y 2 ( c x + a 2 ...

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Author: Rogawski

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Chapter 12.5, Problem 67E

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We need to verify theorem 2 which states that the equation of the hyperbola in standard position is: ${(\frac{x}{a})}^{2} - {(\frac{y}{b})}^{2} = 1$

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Chapter 12.5, Problem 66E

Chapter 12.5, Problem 68E

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Chapter 12 Solutions

CALCULUS (CLOTH)

Ch. 12.1 - Prob. 1PQ Ch. 12.1 - Prob. 2PQ Ch. 12.1 - Prob. 3PQ Ch. 12.1 - Prob. 4PQ Ch. 12.1 - Prob. 5PQ Ch. 12.1 - Prob. 6PQ Ch. 12.1 - Prob. 7PQ Ch. 12.1 - Prob. 1E Ch. 12.1 - Prob. 2E Ch. 12.1 - Prob. 3E

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We need to verify theorem 2 which states that the equation of the hyperbola in standard position is: ( x a ) 2 − ( y b ) 2 = 1

Solution Summary: The author explains that the hyperbola in standard position is: (x a ) 2 (y b ). The plane intersects only one nap of the double right circular cone.

Videos

We need to verify theorem 2 which states that the equation of the hyperbola in standard position is: ${(\frac{x}{a})}^{2} - {(\frac{y}{b})}^{2} = 1$

Want to see the full answer?

Chapter 12 Solutions

We need to verify theorem 2 which states that the equation of the hyperbola in standard position is: ( x a ) 2 − ( y b ) 2 = 1

Solution Summary: The author explains that the hyperbola in standard position is: (x a ) 2 (y b ). The plane intersects only one nap of the double right circular cone.

Videos

We need to verify theorem 2 which states that the equation of the hyperbola in standard position is:(xa)2−(yb)2=1

Want to see the full answer?

Chapter 12 Solutions

We need to verify theorem 2 which states that the equation of the hyperbola in standard position is: ${(\frac{x}{a})}^{2} - {(\frac{y}{b})}^{2} = 1$