Determine the equation of the conic section

[Feodalherren]

Homework Statement

Asymptotes at y=±\sqrt{10} x /5

Foci: (±\sqrt{7},0)

Homework Equations

The Attempt at a Solution

It's obviously a hyperbola. What I can't wrap my head around is why a^2 + b^2 ≠ c^2

c = sqrt 7 ; correct?

This yields
7=25+10
obviously NOT true. What am I missing here?

 

[Mark44]

Homework Statement

Asymptotes at y=±\sqrt{10} x /5

Foci: (±\sqrt{7},0)

Homework Equations

The Attempt at a Solution

It's obviously a hyperbola. What I can't wrap my head around is why a^2 + b^2 ≠ c^2

c = sqrt 7 ; correct?

No. c = √7.

You know that a² + b² = 7.

If one vertex is at (a, 0), use the slope of the asymptote to write b in terms of a, and then solve for a. You should find that a < c.

This yields
7=25+10
obviously NOT true. What am I missing here?

 

[Feodalherren]

Sorry I meant c^2 = 7.

Still. The asymptotes of a hyperbola that shoots of in the x direction is positive and negative a/b.

I don't get this at all.

 

[Mark44]

Sorry I meant c^2 = 7.

Still. The asymptotes of a hyperbola that shoots of in the x direction is positive and negative a/b.

I don't get this at all.

If the hyperbola's vertices are at (±a, 0), the asymptotes are given by y = ±(b/a)x. What are the coordinates of the point on the line y = (b/a)x at which x = a? You are given the slope of the asymptotes, and you also know that a² + b² = 7.

You have two equations that involve a and b, so you should be able to solve for these variables.