The locus of a point P- - moving under the condition that the line y - x - - a tangent to the hyperbola x2-a2 - y2-b2 - 1 is

The correct option is A a hyperbolaIf y=α x+β touches the hyperbola x ^ 2 a ^ 2 y ^ 2 b ^ 2 =1 , then β nbsp; ^ 2 = a ^ 2 α nbsp; ^ 2 ...

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Standard XII

Mathematics

Director Circle: Hyperbola

The locus of ...

Question

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

a hyperbola

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a parabola

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a circle

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an ellipse

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Solution

The correct option is A a hyperbola
If $y = α x + β$ touches the hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$ , then
$β^{2} = a^{2} α^{2} - b^{2}$
Thus locus of $P (α, β)$ is $y^{2} = a^{2} x^{2} - b^{2}$
$a^{2} x^{2} - y^{2} = b^{2}$ which represents a hyperbola
Hence, option 'A' is correct.

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Similar questions

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

Q. The locus of a point

(α, β)

moving under the condition that

y = α x + β

is a tangent to the hyperbola

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

is :

Q. The locus of point of intersection of tangents drawn at

α, β

on the ellipse

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

such that

α - β = \frac{π}{3}

is:

Q. If chord contact of the tangents drawn from the point

(α, β)

to the ellipse

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

touches the circle

x^{2} + y^{2} = c^{2},

THEN THE LOCUS OF THE POINT

(α, β)

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The locus of a point P(α,β) moving under the condition that the line y=αx+β a tangent to the hyperbola x2a2−y2b2=1 is

The correct option is A a hyperbolaIf y=αx+β touches the hyperbola x2a2−y2b2=1, then β2=a2α2−b2Thus locus of P(α,β) is y2=a2x2−b2a2x2−y2=b2 which represents a hyperbolaHence, option 'A' is correct.

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