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

Correct answer: D. a hyperbola Tangent to the hyperbola (x^2)/(a^2)-(y^2)/(b^2)=1 is y=m x ± a^2 m^2-b^2 Given that y=α x+β is the tangent of hyperbola m=α and a^2 m^2-b^2=β^2 a^2 α^2-b^2=β^2 Locus is a^2 x^2-y^2=b^2 which is hyperbola.

JEE Main 2005

Exam: JEE Main
Year: 2005
Subject: Mathematics
Chapter: Hyperbola

JEE Main 2005 · Mathematics · Hyperbola

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

The locus of a point

P(\alpha, \beta)

moving under the condition that the line

y=\alpha x+\beta

is a tangent to the hyperbola

\frac{x^2}{a^2}-\frac{y^2}{b^2}=1

Options

Aan ellipse
Ba circle
Ca parabola
Da hyperbola

Correct answer

D. a hyperbola

Step-by-step solution

Tangent to the hyperbola

\frac{x^2}{a^2}-\frac{y^2}{b^2}=1

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

Given that

y=\alpha x+\beta

is the tangent of hyperbola

\Rightarrow m=\alpha

and

a^2 m^2-b^2=\beta^2

\therefore \mathrm{a}^2 \alpha^2-\mathrm{b}^2=\beta^2

Locus is

a^2 x^2-y^2=b^2

which is hyperbola.

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