Normal are drawn to the hyperbola `(x^2)/(a^2)-(y^2)/(b^2)=1` at point `theta_1a n dtheta_2` meeting the conjugate axis at `G_1a n dG_2,` respectively

Question

Normal are drawn to the hyperbola `(x^2)/(a^2)-(y^2)/(b^2)=1` at point `theta_1a n dtheta_2` meeting the conjugate axis at `G_1a n dG_2,` respectively

Normal are drawn to the hyperbola `(x^2)/(a^2)-(y^2)/(b^2)=1` at point `theta_1a n dtheta_2` meeting the conjugate axis at `G_1a n dG_2,` respectively. If `theta_1+theta_2=pi/2,` prove that `C G_1dotC G_2=(a^2e^4)/(e^2-1)` , where `C` is the center of the hyperbola and `e` is the eccentricity.

Monjur Alahi

4 views

2025-01-04 04:42

1 Answers

Related Questions

From a point `P(1,2)` , two tangents are drawn to a hyperbola `H` in which one tangent is drawn to each arm of the hyperbola. If the equations of the

2 Answers 5 Views

Statement 1 : If from any point `P(x_1, y_1)` on the hyperbola `(x^2)/(a^2)-(y^2)/(b^2)=-1` , tangents are drawn to the hyperbola `(x^2)/(a^2)-(y^2)/(

2 Answers 6 Views

If any line perpendicular to the transverse axis cuts the hyperbola `(x^2)/(a^2)-(y^2)/(b^2)=1` and the conjugate hyperbola `(x^2)/(a^2)-(y^2)/(b^2)=-

2 Answers 6 Views

If the vertex of a hyperbola bisects the distance between its center and the correspoinding focus, then the ratio of the square of its conjugate axis

2 Answers 5 Views

The sides `A Ca n dA B` of a ` A B C` touch the conjugate hyperbola of the hyperbola `(x^2)/(a^2)-(y^2)/(b^2)=1` . If the vertex `A` lies on the ellip

2 Answers 4 Views

The chord of contact of a point `P` w.r.t a hyperbola and its auxiliary circle are at right angle. Then the point `P` lies on conjugate hyperbola one

2 Answers 4 Views

Ifthe normal at P to the rectangular hyperbola `x^2-y^2=4` meets the axes in G and g and C is the centre of the hyperbola, then

2 Answers 4 Views

The locus of the foot of perpendicular from my focus of a hyperbola upon any tangent to the hyperbola is the auxiliary circle of the hyperbola. Consid

2 Answers 6 Views

i. Annual General Meeting, Extra Ordinary General Meeting, Board Meeting. ii. Annual General Meeting, Committee Meeting, Class Meeting.

2 Answers 5 Views

(i) Shareholders Meeting, Directors Meeting, Committee Meeting.

2 Answers 5 Views

Salim Ahmed Kawsar · Accepted Answer · 2023-02-05T15:29:48+06:00

The normal at point `P(a sec theta_(1), b tan theta_(2))` is
`ax cos theta_(1)+by cot theta_(1)=(a^(2)+b^(2))`
It meets the conjugate axis at `G_(1)(0,(a^(2)+b^(2))/(b)tan theta_(1))`.
The normal at point `Q(a sec theta_(2), b tan theta_(2))` is
`axcos theta_(2)+" by " cot theta_(2)=(a^(2)+b^(2))`
It meets the conjugate axis at `G_(1)(0,(a^(2)+b^(2))/(b)tan theta_(2)).` Therefore,
`CG_(1)*CG_(2)=((a^(2+b^(2)))^(2))/(b^(2))tantheta_(1)tantheta_(2)`
`=((a^(2)+b^(2)))/(b^(2))" "(becausetheta_(1)+theta_(2)=(pi)/(2))`
`=(a^(4)(1+(b^(2))/(a^(2))))/(b^(2))`
`=(a^(2)e^(4))/(e^(2)-1)`