Let the coordinate at point of intersection of normals at
P and
Q be
(h,k) Since, equation of normals to the hyperbola
−=1 At point
(x1,y1) is
+=a2+b2 therefore equation of normal to the hyperbola
− =1 at point
P(3secθ,2tanθ) is
+=32+22 ⇒3xcosθ+2ycotθ=32+22 . . . . (1)
Similarly, Equation of normal to the hyperbola
− at point
Q(3secφ,2tanφ) is
+=32+22 ⇒3xcosφ+2ycotφ=32+22 . . . . (2)
Given
θ+φ=⇒φ=−θ and these passes through
(h,k) ∴ From eq. (2)
3xcos(−θ)+2ycot(−θ)=32+22 ⇒3hsinθ+2ktanθ=32+22 . . . . (3)
and
3hcosθ+2kcotθ=32+22 . . . . (4)
Comparing equation (3) & (4), we get
3hcosθ+2kcotθ=3hsinθ+2ktanθ 3hcosθ−3hsinθ=2ktanθ−2kcotθ 3h(cosθ−sinθ)=2k(tanθ−cotθ) 3h(cosθ−sinθ)=2k| (sinθ−cosθ)(sinθ+cosθ) |
| sinθcosθ |
or,
3h=| −2k(sinθ+cosθ) |
| sinθcosθ |
Now, putting the value of equation (5) in eq. (3)
| −2k(sinθ+cosθ)sinθ |
| sinθcosθ |
+2ktanθ=32+22 ⇒2ktanθ−2k+2ktanθ=13 −2k=13⇒k= Hence, ordinate of point of intersection of normals at
P and
Q is