The locus of a point P (α, β) moving under the condition that the line y = + β is a tangent to the hyperbola x^ 2 1 - y^ 2 b^ 2 =1 is a conic, with eccentricity equal to

Correct answer: D. 2 The tangent to the hyperbola x^ 2 1 - y^ 2 b^ 2 = 1 is y = mx ± m^ 2 - b^ 2 Given that, y = + β is a tangent to the hyperbola. So, m = α and m^ 2 - b^ 2 = β ^ 2 ∴ α ^ 2 - b^ 2 = β ^ 2 The locus is x^ 2 - y^ 2 = b^ 2 , which is a hyperbola having eccentricity = 2

NTA Abhyas 2020

Exam: NTA Abhyas
Year: 2020
Subject: Mathematics
Chapter: Hyperbola

NTA Abhyas 2020 · Mathematics · Hyperbola

The locus of a point P (α, β) moving under the condition that the...

The locus of a point

P \left(\alpha, \beta\right) moving under the condition that the line

y = \alphax +

\beta is a tangent to the hyperbola \dfrac{x^{2}}{1} - \dfrac{y^{2}}{b^{2}} =1

is a conic, with eccentricity equal to

Options

A $1$
B $2$
C $\dfrac{1}{2}$
D $\sqrt{2}$

Correct answer

D. $\sqrt{2}$

Step-by-step solution

The tangent to the hyperbola

\dfrac{x^{2}}{1} - \dfrac{y^{2}}{b^{2}} = 1

y = mx \pm \sqrt{m^{2} - b^{2}}

Given that,

y = \alphax + \beta

is a tangent to the hyperbola.
So,

m = \alpha

and

m^{2} - b^{2} = {\beta}^{2}

∴ {\alpha}^{2} - b^{2} = {\beta}^{2}

The locus is

x^{2} - y^{2} = b^{2},

which is a hyperbola having eccentricity

= \sqrt{2}

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