The locus of a point P- - moving under the condition that the line y - - x - - is a tangent to the hyperbola x2-4-y2-9-1 is

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Standard XII

Mathematics

Auxiliary Circle of Hyperbola

The locus of ...

Question

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 $\frac{x^{2}}{4} - \frac{y^{2}}{9} = 1$ is

circle

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ellipse

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parabola

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hyperbola

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Solution

The correct option is D hyperbola
$y = α x + β$ is a tangent
Then $β^{2} = 4 α^{2} - 9$
thus locus of P is $4 x^{2} - y^{2} = 9$ which is hyperbola.

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Similar questions

Q. 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

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. If the tangent at a point

P (α, β)

on the hyperbola

\frac{x^{2}}{25} - \frac{y^{2}}{16} = 1

cuts the circle

x^{2} + y^{2} = 25

at the point

Q (x_{1}, y_{1})

and

R (x_{2}, y_{2}),

then

\frac{y_{1} y_{2}}{y_{1} + y_{2}} =

The equation of chord joining 2 point P $(α)$ and Q $(β)$ in a hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 i s \frac{x}{a} . c o s (\frac{α - β}{2}) - \frac{y}{b} . s i n (\frac{α + β}{2}) = c o s [\frac{α + β}{2}]$ .

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The locus of a point P(α,β) moving under the condition that the line y=αx+β is a tangent to the hyperbola x24−y29=1 is

The correct option is D hyperbolay=αx+β is a tangentThen β2=4α2−9thus locus of P is 4x2−y2=9 which is hyperbola.

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 $\frac{x^{2}}{4} - \frac{y^{2}}{9} = 1$ is

The correct option is D hyperbola
$y = α x + β$ is a tangent
Then $β^{2} = 4 α^{2} - 9$
thus locus of P is $4 x^{2} - y^{2} = 9$ which is hyperbola.