MHT CET 2019 :: Mathematics :: General questions

• Mathematics - General questions

 For a G P, if   $S_{n}=\frac{4^{n}-3^{n}}{3^{n}}$ , then $t_{2}$ = .....

Answer:

Explanation:

We have,  $S_{n}=\frac{4^{n}-3^{n}}{3^{n}}$

 For   $n=1,S_{1}=t_{1}=\frac{4-3}{3}=\frac{1}{3}$

 For   $n=2,S_{2}=t_{1}+t_{2}=\frac{4^{2}-3^{2}}{3^{2}}=\frac{7}{9}$

 $\therefore$      $t_{2}=\frac{7}{9}-t_{1}=\frac{7}{9}-\frac{1}{3}=\frac{7-3}{9}=\frac{4}{9}$

 If   $P( x_{1},y_{1})$  is a point  on the hyperbola  x²-y² =a² , then SP.S'P=.....

Answer:

Explanation:

 Given , equation of hyperbola

    x²-y² =a² ..............(i)

 If P(x₁,y₁) is a point on Eq.(i) , then  

      $x_{1}^{2}-y_{1}^{2}$=$a_{}^{2}$  ......(ii)

 Now,     SP= ex₁-a

     and SP'= ex₁+a

 $\therefore$    SP.SP' = (ex₁-a)(ex₁ +a)

                         = e².x₁²-a²

 = $2x_1^2-a^{2}$  (since , for x²-y²= a², e = $\sqrt{2}$)

  = $2x_1^2-(x_1^2-y_1^2)$    [using Eq. (ii)]

 =  $x_1^2+y_1^2$

• Mathematics - General questions