First slide

Equation of line in Straight lines

Question

A triangle is formed by the lines $x + y = 0$ and $l x + m y = 1$ . If l and m vary subject to the condition $l^{2} + m^{2} = 1$ , then the locus of its circum centre is

Difficult

A

${(x^{2} - y^{2})}^{2} = x^{2} + y^{2}$
B

${(x^{2} + y^{2})}^{2} = (x^{2} - y^{2})$
C

${(x^{2} + y^{2})}^{2} = 4 x^{2} y^{2}$
D

${(x^{2} - y^{2})}^{2} = {(x^{2} + y^{2})}^{2}$

Solution

Coordinates of circum centre are $l / (l^{2} - m^{2}), m / (m^{2} - l^{2})$ .
Hence
$h = \frac{l}{l^{2} - m^{2}} ............... (1)$
$k = - \frac{m}{l^{2} - m^{2}} ............... (2)$
Squaring and adding (1) and (2) we get
$h^{2} + k^{2} = \frac{l^{2} + m^{2}}{{(l^{2} - m^{2})}^{2}} = \frac{1}{{(l^{2} - m^{2})}^{2}}$ (putting $l^{2} + m^{2} = 1$ )
Hence, $\frac{1}{{(l^{2} - m^{2})}^{2}} = {(h^{2} - k^{2})}^{2}$
Therefore, the locus is ${(x^{2} - y^{2})}^{2} = x^{2} + y^{2}$

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