If two tangents can be drawn to distinct branches of hyperbola x2−y24=1 from the point (a,a2), then pick the options1) a∈(−2,0)2) a∈(0,2)3) a∈(−∞,−2)4) a∈(∞,2)

Expert Answer to "If two tangents can be drawn to distinct..."

kadetskihykw

2022-04-23

If two tangents can be drawn to distinct branches of hyperbola

x^{2} - \frac{y^{2}}{4} = 1

from the point

(a, a^{2})

, then pick the options
1)

a \in (- 2, 0)

a \in (0, 2)

a \in (- \infty, - 2)

a \in (\infty, 2)

Aliana Sexton

Beginner2022-04-24Added 20 answers

Note that the asymptotes of

x^{2} - \frac{y^{2}}{4} = 1

are

y = \pm 2 x

, which intersect

y = x^{2}

, the curve of

(a, a^{2})

x = \pm 2

.
Geometrically, for any point

(a, a^{2})

above the asymptotes, there is only one tangent line that can be drawn to either branch of the hyperbola, which can also be seen from that

a^{2} = m a \pm \sqrt{m^{2} - 4}

has only one root m for each branch for

| a | > 2

.
Thus, the options are

a \in (- \infty, - 2)

and

a \in (2, \infty)

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