Question

The polars of point (1,7) (2,6) (t,5) with respect to the circle are concurrent then the value of t is

Answer 1

The polar of point (1,7) (2,6) (t,5) with respect to the circle are concurrent then the value of t is 3.

Condition for concurrency

$\left[\begin{array}{ccc}a&b&c\\d&e&f\\g&h&i\end{array}\right] =0$

here first we need to convert all the point into the line equation.

Since there are 3 coordinate points, so no. of lines formed will be 3.

• A point be(1,7)
• B point be(2,6)
• C point be(t,5)

Then lines will be AB, BC, CA.

$\frac{y-y_{1}}{x-x_{1}}=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}$

AB line will be  $x + y - 8=0$            

BC line will be $x+(t-2)y + (-6t+10)=0$

CA line will be      $2x + (t-1)y+ (-7t+5)=0$

Now put these point in condition of concurrency.

$\left[\begin{array}{ccc}1&1&-8\\1&(t-2)&(-6t+10)\\2&(t-1)&(-7t+5)\end{array}\right] =0$

solving it, $1\left[-7 t^{2}+5 t+14 t-10+6 t^{2}-10 t-6 t+10\right]-1[-7 t+5+12 t-20]+[-8 t+8+16 t-32]=0$

                                                             $-t^{2}+3 t-5 t+15+8 t-24=0$

                                                                                     $-t^{2}+6 t-9=0$

                                                                                       $t^{2}-6 t+9=0$

                                                                                  $t^{2}-3 t-3 t+9=0$

                                                                               $t(t-3)-3(t-3)=0$

                                                                                      $(t-3)(t-3)=0$

                                                                                                           $t=3,3$    

The polar of point (1,7) (2,6) (t,5) with respect to the circle are concurrent then the value of t is 3.

Learn more about Condition for concurrency here,

https://brainly.in/question/63266?msp_poc_exp=1

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