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10-2. Parabola, Ellipse, Hyperbola
normal
The locus of a point $P\left( {\alpha ,\beta } \right)$ 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 hyperbola
B
a parabola
C
a circle
D
an ellipse
Solution
The line $y=m x+c$ is a tangent to the hyperbola
$\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$
if $c^{2}=a^{2} m^{2}-b^{2}$
The line $y=\alpha x+\beta$ is a tangent to
$\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$
So, $\beta^{2}=a^{2} \alpha^{2}-b^{2}$
$\therefore $ Locus of $(\alpha, \beta)$ is
$y^{2}=a^{2} x^{2}-b^{2}$
$\Rightarrow a^{2} x^{2}-y^{2}=b^{2}$
or $\frac{x^{2}}{\left(\frac{b}{a}\right)^{2}}-\frac{y^{2}}{b^{2}}=1,$ which is hyperbola.
Standard 11
Mathematics
