Locus of a point P(α ,,β ) moving under condition that line y = α x + β is a tangent to hyperbol

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10-2. Parabola, Ellipse, Hyperbola

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

A

A parabola

B

A hyperbola

C

An ellipse

D

A circle

(AIEEE-2005)

Solution

(b) If $y = mx + c$ is tangent to the hyperbola then ${c^2} = {a^2}{m^2} – {b^2}$.

Here ${\beta ^2} = {a^2}{\alpha ^2} – {b^2}$.

Hence locus of $P$($\alpha$, $\beta$) is ${a^2}{x^2} – {y^2} = {b^2}$,

which is a hyperbola.

Standard 11

Mathematics

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