Question #223996

The points P(ap2 , 2ap) and Q(aq2 , 2aq) lie on the parabola y2= 4ax. Prove that if PQ is a focal chord then the tangents to the curve at P and Q intersect at right angles at a point on the directrix.


1
Expert's answer
2021-09-20T16:54:15-0400

Solution:

Let's draw a figure as per given data:



Because (P,Q) is focal hord we have:

ap2aq22ap2aq)={a\cdot p^2-a\cdot q^2 \over 2\cdot a\cdot p-2\cdot a\cdot q)}=ap2a2ap;{a\cdot p^2-a\over 2\cdot a\cdot p};

p+q=p1pp+q=p-{1\over p} ;

pq=1p\cdot q=-1

Let us find the intersection of Z(-a,u(p)) tangent at P with asymptota x=-a:

4axy2=0equation ofparabola4\cdot a\cdot x-y^2=0 - equation \space of parabola

2adxydy=0; dx=x(P)x(Z)=ap2+a;dy=y(P)y(Z)=2apu(p);2a(ap2+a)2pa(2apu(p);ap2+a2ap2+pu(p)=0;u(p)=ap2ap=apa1p=a(p+q);2\cdot a\cdot dx-y\cdot dy=0;\space \\ dx=x(P)-x(Z)=a\cdot p^2+a;\\ dy=y(P)-y(Z)=2\cdot a\cdot p-u(p);\\ 2\cdot a\cdot(a\cdot p^2+a)-2\cdot p\cdot a\cdot (2\cdot a\cdot p-u(p); \\a\cdot p^2+a-2\cdot a\cdot p^2+p\cdot u(p)=0;\\ u(p)=\frac{a\cdot p^2-a}{p}=a\cdot p-a\cdot \frac{1}{p}=a\cdot (p+q);

Similarly u(q)=a(q+p)

Thus u(p)=u(q) therefore tangents at P and Q intersecs at Z(-a,a(p+q));

k(P)=dy(p)/dx(P)=2apa(p1p)ap2+a=1p{2\cdot a\cdot p-a\cdot (p-\frac{1}{p})\over a\cdot p^2+a }=\frac{1}{p} is the slope of tangent at P;

Similarly k(Q)=1q\frac{1}{q} .

Therefore k(P)\cdot k(Q)=-1 and tangents at P and at Q are ortogonal in Z.

Hence, proved.


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