Solution:
Let's draw a figure as per given data:
Because (P,Q) is focal hord we have:
2⋅a⋅p−2⋅a⋅q)a⋅p2−a⋅q2=2⋅a⋅pa⋅p2−a;
p+q=p−p1 ;
p⋅q=−1
Let us find the intersection of Z(-a,u(p)) tangent at P with asymptota x=-a:
4⋅a⋅x−y2=0−equation ofparabola
2⋅a⋅dx−y⋅dy=0; dx=x(P)−x(Z)=a⋅p2+a;dy=y(P)−y(Z)=2⋅a⋅p−u(p);2⋅a⋅(a⋅p2+a)−2⋅p⋅a⋅(2⋅a⋅p−u(p);a⋅p2+a−2⋅a⋅p2+p⋅u(p)=0;u(p)=pa⋅p2−a=a⋅p−a⋅p1=a⋅(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)=a⋅p2+a2⋅a⋅p−a⋅(p−p1)=p1 is the slope of tangent at P;
Similarly k(Q)=q1 .
Therefore k(P)⋅ k(Q)=-1 and tangents at P and at Q are ortogonal in Z.
Hence, proved.
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