Multiple choice

If a circle of radius $3$ units touching the lines $\sqrt { 3 } { y }^{ 2 }-4xy+\sqrt { 3 } { x }^{ 2 }=0$ in the first quadrant then the length of the chord of contact to this circle

  1. $\dfrac { \sqrt { 3 } +1 }{ 2 }$
  2. $\dfrac { \sqrt { 3 } +1 }{ \sqrt { 2 } }$
  3. $3\left( \dfrac { \sqrt { 3 } +1 }{ \sqrt { 2 } } \right)$
  4. $\dfrac { \left( \sqrt { 3 } +1 \right) }{ 2 }$
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C Correct answer
Explanation

The given lines represent a pair of straight lines. By simplifying the equation, we find the angle between them and the distance from the origin. The circle touches these lines, and the chord of contact is calculated using the geometry of the circle and the lines.

AI explanation

The given equation of the lines is a homogeneous quadratic representing two lines passing through the origin, and factoring it as (y - square root of 3 * x)(square root of 3 * y - x) = 0 yields slopes showing the angle between the lines is 60 degrees. For a circle of radius 3 inscribed in this 60-degree angle in the first quadrant, the distance from the origin to the center along the angle bisector is found to be 3 times (square root of 3 + 1) divided by the square root of 2. The length of the chord of contact from the origin is found by the formula 2 times the radius multiplied by the length of the tangent, and simplifying this relationship gives the result as 3 times (square root of 3 + 1) divided by the square root of 2.

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