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# If the lines ${a_1}x + {b_1} y + {c_1} = 0$ and ${a_2}x + {b_2} y + {c_2} = 0$ cut the coordinate axis in concyclic points, thenA ${a_1} {a_2} = {b_1} {b_2}$B $\dfrac{{{a_1}}} {{{a_2}}} = \dfrac{{{b_1}}} {{{b_2}}}$C ${a_1} + {a_2} = {b_1} + {b_2}$D ${a_1} {b_1} = {a_2} {b_2}$

Last updated date: 12th Jul 2024
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Hint: In this question we have been given equations of two lines which cut the coordinate axis in concyclic points which gives us a very important relation. Let suppose the first line intersect at the points A and B and similarly the second line at the points C and D. AB and CD are chords around the x and the y axis with origin O. Therefore, the relation between the points A, B, C, D, as they are concyclic would be $OA \times OB = OC \times OD$. We would use this relation to solve the equation further and find the relation between ${a_1}, {a_2}, {b_1}, {b_2}$.

We have been provided with two equation ${a_1}x + {b_1} y + {c_1} = 0$ and ${a_2}x + {b_2} y + {c_2} = 0$. So, we will be finding the coordinates of both the equations one by one.
The first equation is ${a_1}x + {b_1}y + {c_1} = 0$ , so the coordinates of this equation will be $A\left( {\dfrac{{ - {c_1}}}{{{a_1}}},0} \right)$ and $B\left( {0,\dfrac{{ - {c_1}}}{{{b_1}}}} \right)$
Similarly, for the second equation ${a_2}x + {b_2}y + {c_2} = 0$, the coordinates would be $C\left( {\dfrac{{ - {c_2}}}{{{a_2}}},0} \right)$ and $D\left( {0,\dfrac{{ - {c_2}}}{{{b_2}}}} \right)$
And since these points A, B, C, D, are concyclic the following relation would be true for them $OA \times OB = OC \times OD$.
The values will be $\left( {\dfrac{{ - {c_1}}}{{{a_1}}}} \right) \times \dfrac{{ - {c_2}}}{{{a_2}}} = \left( {\dfrac{{ - {c_1}}}{{{b_1}}}} \right) \times \dfrac{{ - {c_2}}}{{{b_2}}}$
Now solving this equation using cross multiplication method the final relation comes out to be ${a_1} {a_2} = {b_1} {b_2}$
So, the answer comes out to be ${a_1} {a_2} = {b_1} {b_2}$ which is your option (a).