1-If-x-2-2-y-2-x-2-2-y-2-6-show-that-when-the-equation-is-simplified-it-can-be-express-as-x-2-9-y-2-5-1-2-find-the-value-of-n-such-that-the-linear-factors-of-the-form-x

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Question Number 53500 by Otchere Abdullai last updated on 22/Jan/19

$$1.If\sqrt{{(x+2)}^2+y^2}+\sqrt{{(x-2)}^2+y^2}=6$$$$showthatwhentheequationis$$$$simplified,itcanbeexpressas$$$$\frac{x^2}{9}+\frac{y^2}{5}=1$$$$2.findthevalueofnsuchthatthe$$$$linearfactorsoftheformx+ay+b$$$$andx+cy+dwithintegercoefficients$$$$havetheproduct{x}^2+5xy+x+ny-n$$$$sirpleasehelp$$

Answered by tanmay.chaudhury50@gmail.com last updated on 22/Jan/19

$$\sqrt{{(x+2)}^2+y^2}=6-\sqrt{{(x-2)}^2+y^2}$$$$x^2+4x+4+y^2=36-12\sqrt{{(x-2)}^2+y^2}+x^2-4x+4+y^2$$$$cancelling{x}^2,y^2,4frombothside$$$$4x=36-12\sqrt{x^2-4x+4+y^2}-4x$$$$8x-36=-12\sqrt{x^2-4x+4+y^2}$$$$2x-9=-3\sqrt{x^2-4x+4+y^2}$$$$4x^2-36x+81=9x^2-36x+36+9y^2$$$$5x^2+9y^2=45$$$$\frac{x^2}{9}+\frac{y^2}{5}=1$$

Commented by Otchere Abdullai last updated on 22/Jan/19

$$wow!thankyousirGodblessyou$$

Answered by tanmay.chaudhury50@gmail.com last updated on 22/Jan/19

$$2)(x+ay+b)(x+cy+d)=x^2+5xy+x+ny-n$$$$x^2+cxy+xd+axy+{acy}^2+ayd+bx+bcy+bd=x^2+5xy+x+ny-n$$$$x^2+xy(a+c)+x(b+d)+y(ad+bc)+{acy}^2+bd=x^2+5xy+x+ny-n$$$$so$$$$a+c=5$$$$b+d=1$$$$ad+bc=n$$$$ac=0$$$$bd=-n$$$$ac=0$$$$a(5-a)=0eithera=0ora=5$$$$whena=0c=5andwhena=5c=0$$$$nowleta=0c.=5$$$$ad+bc=n$$$$0\times d+b\times 5=n[b=\frac{n}{5}]$$$$a=0$$$$b=\frac{n}{5}$$$$c=5$$$$d=1-b=1-\frac{n}{5}$$$$bd=-n$$$$\left(\frac{n}{5}\right)\times (1-\frac{n}{5})=-n$$$$\frac{1}{5}\times \left(\frac{5-n}{5}\right)=-1$$$$5-n=-25$$$$-n=-30\dots n=30$$$$$$

Commented by Otchere Abdullai last updated on 22/Jan/19

$$powerfulMathematician!Wethank$$$$Godforhavingyouinthegroup$$$$Godblessyousir!$$

Commented by tanmay.chaudhury50@gmail.com last updated on 22/Jan/19

$$mathematics\dots activatesonesmindtothink\dots$$

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