JEE MainMathematicsEasy / Moderate/ ToughCircles

Quadrilateral formed by points of intersection of pair of lines
ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0 with coordinate axes is cyclic
if
lehdj.k ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0 }kjk fu:fir

Show answer and solution

y = m_1x + c_1 and y = m_2x + c_2 coordinate axes at A

\left( \frac{- c_{1}}{m_{1}},0 \right),B\left( \frac{- c_{2}}{m_{2}},0 \right)

, C(0, c_1) and D(0, c_2). These points can be concyclic only when
x_1x_2 = y_1y_2
y = m_1 x + c_1 vkSj y = m_2 x + c_2
A

\left( \frac{- c_{1}}{m_{1}},0 \right),B\left( \frac{- c_{2}}{m_{2}},0 \right)

,C(0,
c_1) rFkk D(0, c_2) fcUnq lepØh; gksxsa tc x_1 x_2= y_1y_2

\Rightarrow

\frac{c_{1}c_{2}}{m_{1}m_{2}}

= c_1 c_2

\Rightarrow

m_1 m_2

\Rightarrow

\frac{b}{a}

= 1

Source: Grass-G / M-27 Mathematics Paper 2 15 Jan.docx

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