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STD 11 - 10.1 circle and system of circle — Maths STD 11 Science — Question

CBSE BoardEnglish MediumSTD 11 ScienceMathsSTD 11 - 10.1 circle and system of circle1 Mark
MCQ
Given $\frac{x}{a}\, + \,\frac{y}{b}= 1$ and $ax + by = 1$ are two variable lines, $'a'$ and $'b'$ being the parameters connected by the relation $a^2 + b^2 = ab$. The locus of the point of intersection has the equation
  • ✓
    $x^2 + y^2 + xy - 1 = 0$
  • B
    $x^2 + y^2 - xy + 1 = 0$
  • C
    $x^2 + y^2 + xy + 1 = 0$
  • D
    $x^2 + y^2 - xy - 1 = 0$
✓

Answer

Correct option: A.
$x^2 + y^2 + xy - 1 = 0$
a
Let $(h, k)$ be point of intersection then

$\begin{array}{l}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\frac{h}{a} + \frac{k}{b} =1 \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,given\,\,\,\,{a^2} + {b^2} = ab\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\mathop {ah + kb = 1}\limits_{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_} \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \Rightarrow \frac{a}{b} + \frac{b}{a} = 1\\multiply\,\,\,{h^2} + {k^2} + hk(\frac{b}{a} + \frac{a}{b}) = 1\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{h^2} + {k^2} + hk = 1\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{x^2} + {y^2} + xy - 1 = 0
\end{array}$

Note that the locus is not physically viable 

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