Hyperbola equation

Find the hyperbola equation with the center of S [0; 0], passing through the points:
A [5; 3] B [8; -10]

Result

f = (Correct answer is: f = 7 * x^2 - 3y^2 = 148)

Step-by-step explanation:

$$\begin{gathered} {\displaystyle \frac{(x-x_0{)}^2}{a^2}}-{\displaystyle \frac{(y-y_0{)}^2}{b^2}}=1 \\ {\displaystyle \frac{x^2}{a^2}}-{\displaystyle \frac{y^2}{b^2}}=1 \\ {5}^2\mathrm{/}{a}^2-3^2\mathrm{/}{b}^2=1 \\ {8}^2\mathrm{/}{a}^2-(-10{)}^2\mathrm{/}{b}^2=1 \\ 25\mathrm{/}{a}^2=1+9\mathrm{/}{b}^2 \\ {a}^2=25\mathrm{/}(1+9\mathrm{/}{b}^2) \\ 64\mathrm{/}25\cdot (1+9\mathrm{/}{b}^2)-100\mathrm{/}{b}^2=1 \\ 64\mathrm{/}25\ast (b^2+9)-100=b^2 \\ 64\mathrm{/}25\cdot (b^2+9)-100=b^2 \\ 1.56b^2-76.96=0 \\ p=1.56;q=0;r=-76.96 \\ D=q^2-4pr=0^2-4\cdot 1.56\cdot (-76.96)=480.2304 \\ D>0 \\ {b}_{1,2}={\displaystyle \frac{-q\pm \sqrt{D}}{2p}}={\displaystyle \frac{\pm \sqrt{480.23}}{3.12}} \\ {b}_{1,2}=\pm 7.0237691685685 \\ {b}_1=7.0237691685685 \\ {b}_2=-7.0237691685685 \\ \text{ Factored form of the equation: } \\ 1.56(b-7.0237691685685)(b+7.0237691685685)=0 \\ b=b_1=7.0238\doteq 7.0238 \\ a=\sqrt{25\mathrm{/}(1+9\mathrm{/}{b}^2)}=\sqrt{25\mathrm{/}(1+9\mathrm{/}7.0238^2)}\doteq 4.5981 \\ {x}^2\mathrm{/}(148\mathrm{/}7)-y^2\mathrm{/}(148\mathrm{/}3)=1 \\ f=7\cdot x^2-3y^2=148 \end{gathered}$$

Try another example

Related math problems and questions:

• Right triangle eq2
  Find the lengths of the sides and the angles in the right triangle. Given area S = 210 and perimeter o = 70.
• Hyperbola
  Find the equation of hyperbola that passes through the point M [30; 24] and has focal points at F1 [0; 4 sqrt 6], F2 [0; -4 sqrt 6].
• Two chords
  Calculate the length of chord AB and perpendicular chord BC to circle if AB is 4 cm from the center of the circle and BC 8 cm from the center of the circle.
• Two workers
  Two workers should fulfill certain task together for 5 days. If the first worker increased their performance twice and second twice fell, it took them just four days. For how many days would handle the entire task first worker himself?
• Equation - inverse
  Solve for x: 7: x = 14: 1000
• Sequences AP + GP
  The three numbers that make up the arithmetic sequence have the sum of 30. If we subtract from the first 5, from the second 4 and keep the third, we get the geometric sequence. Find AP and GP members.
• Faces diagonals
  If a cuboid's diagonals are x, y, and z (wall diagonals or three faces), then find the cuboid volume. Solve for x=1.3, y=1, z=1.2
• Two diggers
  Two diggers should dig a ditch. If each of them worked just one-third of the time that the other digger needs, they'd dig up a 13/18 ditch together. Find the ratio of the performance of these two diggers.
• Quadratic function
  Write the equation of the quadratic function, which includes points A (-1, 10), B (2, 19), C (1,4)
• Non linear eqs
  Solve the system of non-linear equations: 3x²-3x-y=-2 -6x²-x-y=-7
• Biquadratic
  By introducing a new variable solve biquadratic equation: - x ⁴ +277 x² -15876=0
• Given 2
  Given g(x)=x²+x+1 where x=t². What is g(t²)?
• Find parameters
  Find parameters of the circle in the plane - coordinates of center and radius: ?
• Sphere from tree points
  Equation of sphere with three point (a,0,0), (0, a,0), (0,0, a) and center lies on plane x+y+z=a
• On line
  On line p: x = 4 + t, y = 3 + 2t, t is R, find point C, which has the same distance from points A [1,2] and B [-1,0].
• A bridge
  A bridge over a river has the shape of the arc with bases of the bridge at the river's edge. At the center of the river, the bridge is 10 feet above the water. At 27 feet from the edge of the river, the bridge is 9 feet above the water. How wide is the ri
• Find the
  Find the number x, which if it increases by 2, then its square increases by 21 percent.