Let A = left{ {left( {x,y} right):,y = mx + 1} right} B = left{ {le

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10-2. Parabola, Ellipse, Hyperbola

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Let $A = \left\{ {\left( {x,y} \right):\,y = mx + 1} \right\}$

$B = \left\{ {\left( {x,y} \right):\,\,{x^2} + 4{y^2} = 1} \right\}$

$C = \left\{ {\left( {\alpha ,\beta } \right):\,\left( {\alpha ,\beta } \right) \in A\,\,and\,\,\left( {\alpha ,\beta } \right) \in B\,\,and\,\alpha \, > 0} \right\}$ .

If set $C$ is singleton set then sum of all possible values of $m$ is

A

$0$

B

$\frac{{\sqrt 3 }}{2}$

C

$ - \frac{{\sqrt 3 }}{2}$

D

none of these

Solution

$\mathrm{y}=\mathrm{mx}+1$ is tangent to ellipse

$\mathrm{x}^{2}+4 \mathrm{y}^{2}=1$ in ${I^{st}}$ quadrant $\therefore \mathrm{m}<0$

$\therefore 1=\mathrm{m}^{2}+\frac{1}{4}$

$m=\frac{\sqrt{3}}{2}$ or $-\frac{\sqrt{3}}{2}$

(reject)

Standard 11

Mathematics

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