Question

Let origin be the centre, $(\pm 3, 0)$ the foci and $\dfrac{3}{2}$ be the eccentricity of a hyperbola. Then the line $2x - y - 1 = 0$

• A. intersects the hyperbola at two points
• B. does not intersect the hyperbola
• C. touches the hyperbola
• D. passes through the vertex of the hyperbola

Hint:

To check intersection, substitute the line into the conic and solve — negative discriminant implies no intersection.

Answer 1

The Correct Option is B

Given $c = 3$ and $e = \dfrac{3}{2} \Rightarrow a = \dfrac{c}{e} = \dfrac{3}{(3/2)} = 2$
So $a^2 = 4$, and $b^2 = c^2 - a^2 = 9 - 4 = 5$
Equation of hyperbola: $\dfrac{x^2}{4} - \dfrac{y^2}{5} = 1$
Substitute $y = 2x - 1$ into hyperbola: $\dfrac{x^2}{4} - \dfrac{(2x - 1)^2}{5} = 1$
Solve and check discriminant. It turns out to be negative → no real roots
Hence, line does not intersect the hyperbola