Question

If the lines $x + ky + 3 = 0$ and $2x - 5y + 7 = 0$ intersect the coordinate axes in the concyclic points, then k=?
A.$\dfrac{{ - 2}}{5}$
B.$\dfrac{{ - 3}}{5}$
C.$\dfrac{3}{2}$
D.$\dfrac{{ - 5}}{3}$

Answer 1

Hint: In this question, we need to determine the value of the constant k present in the linear equation of the line $x + ky + 3 = 0$ such that the lines $x + ky + 3 = 0$ and $2x - 5y + 7 = 0$ intersect the coordinate axes in the concyclic points. For this, we will use the relation between the coefficients of the x and y- axes of the lines which are intersecting at the concyclic points which is given as the product of the coefficient of the x-coordinates in both the equations of the lines is equals to the product of the coefficient of the y-coordinates in both the equations of the lines.

Complete step-by-step answer:
If the two linear equations of the lines \[ax + by + c = 0\] and \[px + qy + r = 0\;\]are intersecting the coordinate axes in the concyclic points then, the product of the coefficient of the x-coordinates in both the equations of the lines is equals to the product of the coefficient of the y-coordinates in both the equations of the lines. Mathematically, \[ap = bq\]( for the lines \[ax + by + c = 0\] and \[px + qy + r = 0\;\]).
Comparing the given linear equations of the lines $x + ky + 3 = 0$ and $2x - 5y + 7 = 0$ with the standard linear equation of the lines \[ax + by + c = 0\] and \[px + qy + r = 0\;\] to determine the coefficient of x and y.
$
  ax + by + c = x + ky + 3 \\
  a = 1{\text{ and, }}b = k - - - - (i) \\
 $
Similarly,
$
  px + qy + r = 2x - 5y + 7 \\
  p = 2{\text{ and, }}q = - 5 - - - - (ii) \\
 $

Now, substituting the values of $a = 1{\text{, }}b = k,p = 2{\text{ and, }}q = - 5$ in the equation \[ap = bq\] to determine the value of the constant k.
\[
  ap = bq \\
  1 \times 2 = k \times ( - 5) \\
  k = \dfrac{{ - 2}}{5} \\
 \]
Hence, the value of k in the linear equation of the line $x + ky + 3 = 0$ such that $x + ky + 3 = 0$ and $2x - 5y + 7 = 0$ intersect the coordinate axes in the concyclic points is \[\dfrac{{ - 2}}{5}\].
So, the correct answer is “Option A”.

Note: It is worth noting down here that the concyclic points are the points which lie on the same circle. Here the lines are intersecting at the points which are lying on the circumference of the circle.