First slide

Definition of a circle

Question

Let the lines $(y - 2) = m_{1} (x - 5) and (y + 4) = m_{2} (x - 3)$ intersect at right angles at P (where m₁ and m₂are parameters). If the locus of P is $x^{2} + y^{2} + gx + fy + 7 = 0,$ then the value of f+ g is ______.

Moderate

Solution

Clearly, the locus of the point of intersection of lines is $(x - 5) (x - 3) + (y - 2) (y + 4) = 0$
or $x^{2} + y^{2} - 8 x + 2 y + 7 = 0$
Hence, $f + g = - 8 + 2 = - 6$

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