Answer to Question #124544 in Geometry for jim

Question #124544
1. (a) If a chord of the parabola y^2 = 4ax is a normal at one of its ends, show that its mid-point lies on the curve 2(x − 2a) = y^2/a + 8a^3/y^2
.
Prove that the shortest length of such a chord is 6a√3.

(b) Find the asymptotes of the hyperbola
x^2 − y^2 + 2x + y + 9 = 0.
1
Expert's answer
2020-07-02T19:04:54-0400

If the normal at the point 

P(at12,2at1)P(at_1^2 ,2at_1 ) meets the parabola  y2=4axy^2 =4ax  again at Q(at22,2at2)Q(at^2_2 ,2at^2 ​ ) then

t2​=−t1-2/t1

Q=(a(t1+2t1)2,2a(t1+2t1))Q=(a(t_1+{2\over t_1})^2, -2a(t_1+{2\over t_1}))

Let M(x,y) be the midpoint of PQ

(x,y)=(at12+a(t1+2t1)22,2at12a(t1+2t1)2)(x, y)=\big(\dfrac{at_1^2+a(t_1+\dfrac{2}{t_1})^2}{2}, \dfrac{2at_1-2a(t_1+\dfrac{2}{t_1})}{2}\big)

t1=2ayt_1=-\dfrac{2a}{y}


x=at12+2a+2at12x=at_1^2+2a+\dfrac{2a}{t_1^2}

x2a=4a3y2+y22ax-2a=\dfrac{4a^3}{y^2}+\dfrac{y^2}{2a}


2(x2a)=y2a+8a3y22(x-2a)=\dfrac{y^2}{a}+\dfrac{8a^3}{y^2}


PQ2=(at12+a(t1+2t1)2)2+(2a(t1+2t1)2at1)2PQ^2=(-at_1^2+a(t_1+\dfrac{2}{t_1})^2)^2+(-2a(t_1+{2\over t_1})-2at_1)^2


PQ2=(4a+4at12)2+(4at1+4at1)2PQ^2=(4a+\dfrac{4a}{t_1^2})^2+(4at_1+\dfrac{4a}{t_1})^2

PQ2=16a2(1+2t12+1t12+t12+2+1t14)PQ^2=16a^2(1+\dfrac{2}{t_1^2}+\dfrac{1}{t_1^2}+t_1^2+2+\dfrac{1}{t_1^4})

PQ2=16a2(t12+3t12+1t14+3)PQ^2=16a^2(t_1^2+\dfrac{3}{t_1^2}+\dfrac{1}{t_1^4}+3)


Consider the function


f(u)=u+3u+1u2+3,u>0;f(u)=13u22u3;f(u)=0=>u33u2u3=0;u2(u2)+2u(u2)+(u2)=0;(u2)(u2+2u+1)=0;(u2)(u+1)2=0f(u)=u+\dfrac{3}{u}+\dfrac{1}{u^2}+3, u>0; f'(u)=1-\dfrac{3}{u^2}-\dfrac{2}{u^3} ; f'(u)=0=>\dfrac{u^3-3u-2}{u^3}=0; u^2(u-2)+2u(u-2)+(u-2)=0; (u-2)(u^2+2u+1)=0; (u-2)(u+1)^2=0

Since u>0,  we take u=2

0<u<2,f′(u)<0,f(udecreases

u>2,f′(u)>0,f(uincreases

f(2)=2+32+14+3=274f(2)=2+\dfrac{3}{2}+\dfrac{1}{4}+3=\dfrac{27}{4}

PQ216a2(274)PQ^2\geq16a^2(\dfrac{27}{4})

PQ2108a2PQ^2\geq108a^2

PQ63a,a>0PQ\geq6\sqrt{3}a, a>0

b) A hyperbola with a horizontal transverse axis and center at (h, k) has one asymptote with equation y = k + a(x - h) and the other with equation y = k - a(x - h).a = distance from vertices to the centery=332332(x(1))+12,y=332332(x(1))+12y=\frac{\frac{\sqrt{33}}{2}}{\frac{\sqrt{33}}{2}}\left(x-\left(-1\right)\right)+\frac{1}{2},\:\quad \:y=-\frac{\frac{\sqrt{33}}{2}}{\frac{\sqrt{33}}{2}}\left(x-\left(-1\right)\right)+\frac{1}{2}

y=x+1+12,y=(x+1)+12y=x+1+\frac{1}{2},\:\quad \:y=-\left(x+1\right)+\frac{1}{2}




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