if-point-of-intersection-of-curves-C-1-x-2-4y-2-2xy-9x-3-and-C-2-2x-2-3y-2-4xy-3x-1-subtends-a-right-angle-at-origin-the-value-of-is-

Question Number 31147 by momo last updated on 03/Mar/18

if point of intersection of curves C_1 =λx^2 +4y^2 −2xy−9x+3 and C_2 =2x^2 +3y^2 −4xy+3x−1 subtends a right angle at origin the value of λ is?

$${if}\:{point}\:{of}\:{intersection}\:{of}\:{curves} \\ $$$${C}_{\mathrm{1}} =\lambda{x}^{\mathrm{2}} +\mathrm{4}{y}^{\mathrm{2}} −\mathrm{2}{xy}−\mathrm{9}{x}+\mathrm{3}\:{and} \\ $$$${C}_{\mathrm{2}} =\mathrm{2}{x}^{\mathrm{2}} +\mathrm{3}{y}^{\mathrm{2}} −\mathrm{4}{xy}+\mathrm{3}{x}−\mathrm{1}\: \\ $$$${subtends}\:{a}\:{right}\:{angle}\:{at}\:{origin}\:{the} \\ $$$${value}\:{of}\:\lambda\:{is}? \\ $$

Answered by ajfour last updated on 03/Mar/18

(y_1 /x_1 )=m_1 and (y_2 /x_2 ) =m_2 (say) m_1 m_2 =−1 As points of intersection say A(x_1 ,y_1 ) and B(x_2 ,y_2 ) lies on both curves C_1 +3C_2 gives (λ+6)x^2 +13y^2 −14xy = 0 ⇒ (λ+6)+13((y/x))^2 −14((y/x))=0 as m_1 , m_2 are roots of this eq. m_1 m_2 = ((λ+6)/(13)) = −1 ⇒ 𝛌 = −19 .

$$\:\:\:\:\:\frac{{y}_{\mathrm{1}} }{{x}_{\mathrm{1}} }={m}_{\mathrm{1}} \:\:\:{and}\:\:\:\frac{{y}_{\mathrm{2}} }{{x}_{\mathrm{2}} }\:={m}_{\mathrm{2}} \:\left({say}\right) \\ $$$${m}_{\mathrm{1}} {m}_{\mathrm{2}} =−\mathrm{1} \\ $$$${As}\:{points}\:{of}\:{intersection}\:{say} \\ $$$${A}\left({x}_{\mathrm{1}} ,{y}_{\mathrm{1}} \right)\:{and}\:{B}\left({x}_{\mathrm{2}} ,{y}_{\mathrm{2}} \right)\:{lies}\:{on} \\ $$$${both}\:{curves} \\ $$$${C}_{\mathrm{1}} +\mathrm{3}{C}_{\mathrm{2}} \:{gives} \\ $$$$\left(\lambda+\mathrm{6}\right){x}^{\mathrm{2}} +\mathrm{13}{y}^{\mathrm{2}} −\mathrm{14}{xy}\:=\:\mathrm{0} \\ $$$$\Rightarrow\:\left(\lambda+\mathrm{6}\right)+\mathrm{13}\left(\frac{{y}}{{x}}\right)^{\mathrm{2}} −\mathrm{14}\left(\frac{{y}}{{x}}\right)=\mathrm{0} \\ $$$${as}\:\:{m}_{\mathrm{1}} ,\:{m}_{\mathrm{2}} \:\:{are}\:{roots}\:{of}\:{this}\:{eq}. \\ $$$$\:\:\:{m}_{\mathrm{1}} {m}_{\mathrm{2}} =\:\frac{\lambda+\mathrm{6}}{\mathrm{13}}\:=\:−\mathrm{1} \\ $$$$\Rightarrow\:\:\boldsymbol{\lambda}\:=\:−\mathrm{19}\:\:. \\ $$

Commented by momo last updated on 03/Mar/18