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Question-199392




Question Number 199392 by Calculusboy last updated on 03/Nov/23
Answered by mr W last updated on 03/Nov/23
(x−3)^2 +(y+4)^2 =53  (x+2)^2 +(y−1)^2 =13    eqn. of intersection line:  (x−3)^2 −(x+2)^2 +(y+4)^2 −(y−1)^2 =53−13  −5(2x−1)+5(2y+3)=40  −x+y=2  ⇒y=x+2    intersection points:  (x−3)^2 +(x+2+4)^2 =53  x^2 +3x−4=0  (x−1)(x+4)=0  ⇒x_1 =−4, x_2 =1  ⇒y_1 =−2, y_2 =3  i.e. intersection at (−4,−2) and (1,3)
$$\left({x}−\mathrm{3}\right)^{\mathrm{2}} +\left({y}+\mathrm{4}\right)^{\mathrm{2}} =\mathrm{53} \\ $$$$\left({x}+\mathrm{2}\right)^{\mathrm{2}} +\left({y}−\mathrm{1}\right)^{\mathrm{2}} =\mathrm{13} \\ $$$$ \\ $$$${eqn}.\:{of}\:{intersection}\:{line}: \\ $$$$\left({x}−\mathrm{3}\right)^{\mathrm{2}} −\left({x}+\mathrm{2}\right)^{\mathrm{2}} +\left({y}+\mathrm{4}\right)^{\mathrm{2}} −\left({y}−\mathrm{1}\right)^{\mathrm{2}} =\mathrm{53}−\mathrm{13} \\ $$$$−\mathrm{5}\left(\mathrm{2}{x}−\mathrm{1}\right)+\mathrm{5}\left(\mathrm{2}{y}+\mathrm{3}\right)=\mathrm{40} \\ $$$$−{x}+{y}=\mathrm{2} \\ $$$$\Rightarrow{y}={x}+\mathrm{2} \\ $$$$ \\ $$$${intersection}\:{points}: \\ $$$$\left({x}−\mathrm{3}\right)^{\mathrm{2}} +\left({x}+\mathrm{2}+\mathrm{4}\right)^{\mathrm{2}} =\mathrm{53} \\ $$$${x}^{\mathrm{2}} +\mathrm{3}{x}−\mathrm{4}=\mathrm{0} \\ $$$$\left({x}−\mathrm{1}\right)\left({x}+\mathrm{4}\right)=\mathrm{0} \\ $$$$\Rightarrow{x}_{\mathrm{1}} =−\mathrm{4},\:{x}_{\mathrm{2}} =\mathrm{1} \\ $$$$\Rightarrow{y}_{\mathrm{1}} =−\mathrm{2},\:{y}_{\mathrm{2}} =\mathrm{3} \\ $$$${i}.{e}.\:{intersection}\:{at}\:\left(−\mathrm{4},−\mathrm{2}\right)\:{and}\:\left(\mathrm{1},\mathrm{3}\right) \\ $$
Commented by Calculusboy last updated on 03/Nov/23
thanks sir
$$\boldsymbol{{thanks}}\:\boldsymbol{{sir}} \\ $$

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