The length of chord of contact of the point $( 3,6 )$ with respect to the circle $x ^ { 2 } + y ^ { 2 } = 10$ is
- $\dfrac { 2 \sqrt { 70 } } { 3 }$
- $6 \sqrt { 5 }$
- $\sqrt { 5 }$
- $\dfrac { 12 } { \sqrt { 5 } }$
Chord of contact for (x1, y1) with circle x^2+y^2=r^2 is xx1+yy1=r^2. Here 3x+6y=10. Distance from center (0,0) to chord = |0+0-10| / sqrt(3^2+6^2) = 10 / sqrt(45) = 10 / 3sqrt(5) = 2sqrt(5)/3. Length of chord = 2 * sqrt(r^2 - d^2) = 2 * sqrt(10 - 20/9) = 2 * sqrt(70/9) = 2sqrt(70)/3.
For the circle x squared plus y squared equals 10, the radius r is the square root of 10 and the center O is (0,0). The length of the tangent from the external point P (3, 6) is l equals the square root of (3 squared plus 6 squared minus 10), which equals the square root of 35. The distance d from the center O to P is the square root of (3 squared plus 6 squared), which equals the square root of 45 or 3 times the square root of 5. Using the formula for the length of the chord of contact, 2 times l times r squared divided by d, we get 2 multiplied by the square root of 35 multiplied by 10, all divided by 3 times the square root of 5, which simplifies to 2 times the square root of 70 divided by 3.