Question

Tangents are drawn from point on line $ x - y = 5 $ to $ {x^2} + 4{y^2} = 4 $ . Then all chord of contact pass through a fixed point, whose co-ordinate are:
 $
  A.\,\,\left( {\dfrac{4}{5}, - \dfrac{1}{5}} \right) \\
  B.\,\,\left( {\dfrac{1}{5}, - \dfrac{4}{5}} \right) \\
  C.\,\,\left( {\dfrac{4}{5},\dfrac{1}{5}} \right) \\
  D.\,\,\left( { - \dfrac{4}{5},\dfrac{1}{5}} \right) \\
  $

Answer 1

Hint: To find required solution we first let $ P(a,\,b) $ any point on given line as this point line on line so it will satisfy it and forming first equation and then forming equation of chord of contact with respect to taken point P(a, b) and then simplifying both equations together to find required point or solution of given problem.

Complete step-by-step answer:
Let $ P\left( {a,\,b} \right) $ be any point on line $ x - y = 5 $ .

Since, point $ P\left( {a,\,b} \right) $ lies on line $ x - y = 5 $ . Therefore, it will satisfy the line.
Therefore, substituting point $ P\left( {a,\,b} \right) $ in line. We have,
 $
  a - b = 5 \\
  or \\
  a = 5 + b................(i) \;
  $
Since, it is given that tangents are drawn from $ P\left( {a,\,b} \right) $ to $ {x^2} + 4{y^2} = 4 $
Hence, equation of chord of contact to equation will be given as:
 $ x{x_1} + 4y{y_1} = 4 $
Taking $ \left( {{x_1},{y_1}} \right)\,\,as\,\left( {a,\,b} \right) $ in above equation. We have
 $ ax + 4yb = 4 $ ………….(ii)
Substituting value of ‘a’ from equation (i) in (ii). We have,
 $
  \left( {5 + b} \right)x + 4yb = 4 \\
   \Rightarrow 5x + bx + 4by - 4 = 0 \\
   \Rightarrow \left( {5x - 4} \right) + b\left( {x + 4y} \right) = 0 \;
  $
Above equation of a line is from $ P + \lambda Q = 0 $ family of lines. To find the point of intersection we will equate both P and Q to zero.
Therefore, we have:
 $
  5x - 4 = 0 \\
   \Rightarrow x = \dfrac{4}{5} \;
  $
And
  $
  x + 4y = 0 \\
   \Rightarrow y = - \dfrac{x}{4} \;
  $
Using value of x calculated above in the above equation to find value of y.
 $
  y = - \dfrac{1}{4} \times \dfrac{4}{5} \\
   \Rightarrow y = - \dfrac{1}{5} \\
  $
Hence, the chord of contact passes through a fixed point. Which is given as: $ \left( {\dfrac{4}{5},\,\dfrac{{ - 1}}{5}} \right) $
So, the correct answer is “Option A”.

Note: In this type of problem we first consider any point on given line then using this point we will form an equation of chord or contact to given curve or equation of an ellipse and also point will satisfy given line and then solving both equation so formed to have an equation of family of line of the form $ P + \lambda Q = 0 $ . From which on equating $ P = 0\,\,and\,\,Q = 0 $ we will get the required point through which chord of contact passes through.