Home
Class 12
MATHS
If the normal at P(asectheta,btantheta) ...

If the normal at `P(asectheta,btantheta)` to the hyperbola `x^2/a^2-y^2/b^2=1` meets the transverse axis in G then minimum length of PG is

A

`(b^(2))/(a)`

B

`|(a)/(b)(a+b)|`

C

`|(a)/(b)(a-b)|`

D

`|(a)/(b)(a-b)|`

Text Solution

Verified by Experts

The correct Answer is:
A
Equation of the normal is `ax cos theta + by cot theta = a^(2) +b^(2)` The normal at P meets the coordinate axes at `G((a^(2)+b^(2))/(a)sec theta,0)`
and `g (0,(a^(2)+b^(2))/(b)tan theta)`
`:. PG^(2) = ((a^(2)+b^(2))/(a)sec theta -a sec theta)^(2)+(b tan theta -0)^(2)`
`PG^(2) =(b^(2))/(a^(2)) (b^(2) sec^(2) theta + a^(2) tan^(2) theta)`
When `tan theta =0`
`PG =(b^(2))/(a)`
Promotional Banner

Similar Questions

Explore conceptually related problems

If the normal at P(a sec theta,b tan theta) to the hyperbola (x^(2))/(a^(2))-(y^(2))/(b^(2))=1 meets the transverse axis in G then minimum length of PG is

If the normal at 'theta' on the hyperbola (x^(2))/(a^(2))-(y^(2))/(b^(2))=1 meets the transverse axis at G , and A and A' are the vertices of the hyperbola , then AC.A'G=

If the tangents at the point (a sec alpha, b tan alpha) to the hyperbola (x^(2))/(a^(2))-(y^(2))/(b^(2))=1 meets the transverse axis at T , then the distance of T from a focus of the hyperbola, is

If the normal at P(theta) on the hyperbola (x^(2))/(a^(2))-(y^(2))/(2a^(2))=1 meets the transvers axis at G, then prove that AG*A'G=a^(2)(e^(4)sec^(2)theta-1) where A and A' are the vertices of the hyperbola.

If the normal at P(theta) on the ellipse x^(2)/a^(2)+y^(2)/b^(2)=1 with focus S ,meets the major axis in G. then SG=

If the tangent at the point (asec alpha, b tanalpha ) to the hyberbola (x^(2))/(a^(2)) -(y^(2))/(b^(2)) =1 meets the transverse axis at T. Then the distances of T form a focus of the hyperbola is

If the normal at a pont P to the hyperbola x^2/a^2 - y^2/b^2 =1 meets the x-axis at G , show that the SG = eSP.S being the focus of the hyperbola.

x=a sin theta , y=btantheta so prove that a^2/x^2-b^2/y^2=1

If the normals at P(theta) and Q((pi)/(2)+theta) to the ellipse (x^(2))/(a^(2))+(y^(2))/(b^(2))=1 meet the major axis at G and g, respectively,then PG^(2)+Qg^(2)=b^(2)(1-e^(2))(2-e)^(2)a^(2)(e^(4)-e^(2)+2)a^(2)(1+e^(2))(2+e^(2))b^(2)(1+e^(2))(2+e^(2))

If the normal at any point P on the ellipse x^2/64+y^2/36=1 meets the major axis at G_1 and the minor axis at G_2 then the ratio of PG_1 and PG_2 is equal to