Question

The locus of the mid-points of the chords of the circle \[{x^2} + {y^2} = 16\;\] which are tangents to the hyperbola \[9{x^2} - 16{y^2} = 144\;\] is :
A. \[{\left( {{x^2} + {y^2}} \right)^2} = 16{x^2} - 9{y^2}\]
B. \[{\left( {{x^2} + {y^2}} \right)^2} = 9{x^2} - 16{y^2}\]
C. \[{\left( {{x^2} - {y^2}} \right)^2} = 16{x^2} - 9{y^2}\]
D.None of these

Answer 1

Hint: To get the locus of mid points mid points of the chord of contact first suppose an unknown point as the mid point then just write the equation of chord of contact of circle then apply the condition of tangency on hyperbola.

Complete step-by-step answer:
Let \[P\left( {{x_1},{y_1}} \right)\;\] be the mid point of a chord of the circle.
The equation of the chord is \[T = {S_1}\]
On putting the coordinates of the mid point we get the equation of chord
 \[
   \Rightarrow x{x_1} + y{y_1} - 16 = {x_1}^2 + {y_1}^2 - 16 \\
  y = - \dfrac{{{x_1}}}{{{y_1}}}x + \dfrac{{{x_1}{^2} + {y_{1}}^2}}{{{y_1}}} \\
 \]
Since it chord touches the hyperbola \[\dfrac{{{x^2}}}{{{4^2}}} - \dfrac{{{y^2}}}{{{3^2}}} = 1\]
∴ \[{\left( {\dfrac{{{x_1}^{2} + {y_1}{^2}}}{{{y_1}}}} \right)^2} = {4^2}.\dfrac{{{x_{1}}^{2}}}{{{y_1}^2}} - {3^2}\;\;\;\;\;\;\;\;\;\;\] \[[\because {c^2} = {a^2}{m^2} - {b^2}] \]
∴ Locus of \[P\left( {{x_1},{y_1}} \right)\;\] is
 \[
  \dfrac{{{{\left( {{x^2} + {y^2}} \right)}^2}}}{{{y^2}}} = \dfrac{{16{x^2} - 9{y^2}}}{{{y^2}}} \\
 \]
Or on cancelling the like terms
 \[{\left( {{x^2} + {y^2}} \right)^2} = 16{x^2} - 9{y^2}\]
Hence the locus of mid point of the chord of the circle which is tangent to the hyperbola is
 \[{\left( {{x^2} + {y^2}} \right)^2} = 16{x^2} - 9{y^2}\]
Therefore the option A is the correct answer for this question.
So, the correct answer is “Option A”.

Note: In this type of question where it is asked to find the locus of something, follow all the given conditions accordingly to the question then we will get the required equation of locus. Here we equate the equation of the midpoint of the chord of circle and tangent to the hyperbola to find the assumed variable.