Solution Section 6 Page 55 Math Study Topic 10 – Kite>

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