Let $$E1$$ and $$E2$$ be two ellipses whose centers are at the origin. The major axes of $$E1$$ and $$E2$$ lie along the $$x-axis$$ and the $$y-axis$$, respectively. Let S be the circle $$x2+y$$−$$12=2$$. The straight line $$x+y=3$$ touches the curves S, $$E1$$ and $$E2$$ at P, Q and R, respectively. Suppose that $$PQ=PR=223$$. If $$e1and$$ $$e2are$$ the eccentricities of $$E1$$ and $$E2$$ respectively, then the correct $$expression(s)$$ is (are)

Question

Let $$E1$$ and $$E2$$ be two ellipses whose centers are at the origin. The major axes of   $$E1$$ and $$E2$$  lie along the $$x-axis$$ and the $$y-axis$$, respectively. Let S be the circle $$x2+y$$−$$12=2$$. The straight line $$x+y=3$$ touches the curves S,  $$E1$$ and $$E2$$ at P, Q and R, respectively. Suppose that $$PQ=PR=223$$. If  $$e1and$$  $$e2are$$ the eccentricities of   $$E1$$ and $$E2$$   respectively, then the correct $$expression(s)$$ is (are)

Infinity Learn · Accepted Answer

The line $$x+y=3$$ touches the circle $$x2+y$$−$$12=2$$ at PP is foot of perpendicular from $$0$$,$$1$$ to $$x+y=3$$⇒ $$P1$$,$$2Since$$ $$PQ=223=PR$$  we have $$Q1+2232$$, $$2$$−$$2232E1$$:$$x2a2+y2b2=1$$ with $$e12=a2$$−$$b2a2$$ ∴  $$a2+b2=9$$  $$x+y=3$$  touches  $$E1$$⇒$$a2m2+b2=c2Since$$ Q lies on $$E1$$, we have $$259a2+169b2=1$$⇒ $$a2=5$$, $$b2=4$$∴$$e12=15$$ similarly $$R1$$−$$23$$,  $$2+23$$ lies on $$x2c2+y2d2=1$$   c

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