If the line $\mathrm{x}-1=0$, is a directrix of the hyperbola ${\mathrm{kx}}^2-{\mathrm{y}}^2=6$, then the hyperbola passes through the point

Complete Solution:

The equation of the hyperbola is given by: kx² - y² = 6.

The directrix of a hyperbola is related to its geometry. The given directrix x - 1 = 0 means that the directrix is located at x = 1.

We are given that the point (√5, -2) lies on the hyperbola. Substitute this point into the equation to determine the value of k:

Substitute x = √5 and y = -2:

k(√5)² - (-2)² = 6

5k - 4 = 6

5k = 10

k = 2

Now that we have k = 2, substitute back into the hyperbola equation:

2x² - y² = 6

To verify, substitute the point (√5, -2) again:

2(√5)² - (-2)² = 6

2 × 5 - 4 = 6

10 - 4 = 6

6 = 6

Final Answer:

The hyperbola passes through the point (√5, -2).