Home
Class 12
MATHS
If normal at any point P to ellipse (x^...

If normal at any point P to ellipse `(x^2)/(a^2) + (y^2)/(b^2) = 1(a > b)` meet the x & y axes at A and B respectively. Such that `(PA)/(PB) = 3/4` , then eccentricity of the ellipse is:

A

`1/3`

B

`1/2`

C

`1/4`

D

`1/9`

Text Solution

Verified by Experts

The correct Answer is:
B
Equation of normal to ellipse
`(ax)/(cos theta)==-(by)/(sin theta)=a^(2)-b^(2)`
For point A
`y=0`
`x=cos theta((a^(2)-b^(2))/(a))`

For point B
`x=0`
`y=sin theta ((b^(2)-a^(2))/(3))`

`(a cos theta, b cos theta)(cos theta((a^(2)-b^(2))/(a)),0)(0, sin theta((b^(2)-a^(2))/(b)))`
`0=(b sin thetaxx1+3sin theta((b^(2)-a^(2))/(b)))/(4)`
`b^(2)+3(b^(2)-a^(2))=0`
`4b^(2)-3a^(2)=0 rarr (b^(2))/(a^(2))=(3)/(4)rarr" can be directly used"`
`e=sqrt(1-(3)/(4))=(1)/(2)`
Promotional Banner

Topper's Solved these Questions

  • MOCK TEST 2

    VMC MODULES ENGLISH|Exercise PART III : MATHEMATICS (SECTION - 2)|10 Videos

  • MOCK TEST 13

    VMC MODULES ENGLISH|Exercise MATHEMATICS( SECTION-2)|5 Videos

  • MOCK TEST 3

    VMC MODULES ENGLISH|Exercise PART III : MATHEMATICS (SECTION-2)|10 Videos

Similar Questions

Explore conceptually related problems

If normal at any poin P on the ellipse (x^(2))/(a^(2))+(y^(2))/(b^(2))=1 (agtbgt0) meets the major and minor axes at Q and R, respectively, so that 3PQ=2PR, then find the eccentricity of ellipse

If the normal at point P (theta) on the ellipse x^2/a^2 + y^2/b^2 = 1 meets the axes of x and y at M and N respectively, show that PM : PN = b^2 : a^2 .

If the normal at any point P of the ellipse (x^(2))/(16)+(y^(2))/(9) =1 meets the coordinate axes at M and N respectively, then |PM|: |PN| equals

If the normal at any point P of the ellipse x^2/a^2 + y^2/b^2 = 1 meets the major and minor axes at G and E respectively, and if CF is perpendicular upon this normal from the centre C of the ellipse, show that PF.PG=b^2 and PF.PE=a^2 .

If the normal at any point P on ellipse (x^(2))/(a^(2))+(y^(2))/(b^(2)) =1 meets the auxiliary circle at Q and R such that /_QOR = 90^(@) where O is centre of ellipse, then

If e' is the eccentricity of the ellipse (x^(2))/(a^(2)) + (y^(2))/(b^(2)) =1 (a gt b) , then

The tangent and normal at any point P of an ellipse x^(2)/a^(2)+y^(2)/b^(2)=1 cut its major axis in point Q and R respectively. If QR=a prove that the eccentric angle of the point P is given by e^(2)cos^(2)phi+cosphi-1=0

If the normal at any point P on the ellipse x^2/a^2+y^2/b^2=1 meets the axes at G and g respectively, then find the ratio PG:Pg . (a) a : b (b) a^2 : b^2 (c) b : a (d) b^2 : a^2

A normal inclined at 45^@ to the axis of the ellipse x^2 /a^2 + y^2 / b^2 = 1 is drawn. It meets the x-axis & the y-axis in P & Q respectively. If C is the centre of the ellipse, show that the area of triangle CPQ is (a^2 - b^2)^2/(2(a^2 +b^2)) sq units

If e is eccentricity of the ellipse (x^(2))/(a^(2))+(y^(2))/(b^(2))=1 (where,a lt b), then