** Practice**

Let the hyperbola (H) have a vertex of \({A_1}( – 4;0)\) and a focal length of 10. Write the canonical equation and draw the hyperbola (H).

**Detailed explanation:**

Hyperbola (H) has a vertex \({A_1}( – a;0) = ( – 4;0) \Rightarrow a = 4\)

Focal length \(2c = 10 \Rightarrow c = 5 \Rightarrow b = \sqrt {{c^2} – {a^2}} = \sqrt {{5^2} – {4^2}} = 3\ )

The canonical equation for the hyperbola is: \(\frac{{{x^2}}}{{{4^2}}} – \frac{{{y^2}}}{{{3^2}} } = 1\)

* Draw hyperbola

Step 1: Draw a basic rectangle with four sides belonging to 4 lines \(x = – 4,x = 4,y = 3,y = – 3\)

Step 2: Draw two diagonals of the base rectangle. The point \(M(\frac{{20}}{3};4)\) belongs to (H). Hence the points \({M_1}(\frac{{20}}{3}; – 4),{M_2}( – \frac{{20}}{3};4),{M_3}( – \ frac{{20}}{3}; – 4)\) belongs to (H).

Step 3: Draw the hyperbola