Home
Class 12
MATHS
If the straight line 4ax+3by=24 is a no...

If the straight line `4ax+3by=24` is a normal to the ellipse `(x^(2))/(a^(2))+(y^(2))/(b^(2))=1 (agtb)`, then find the the coordinates of focii

Text Solution

Verified by Experts

The correct Answer is:
`(4+-sqrt(10),0)`
Given normal is `4ax+3by=34 " "(1)`
Let this line be normal to the ellipse at `P(a cos theta, b sin theta)` is `ax sec theta-"by cosec" theta= a^(2)-b^(2)" "(2)`
Equations (1) and (2) are identical, so `(sectheta)/(4)=(-"cosec"theta)/(3)=(a^(2)-b^(2))/(24)`
`:. cos theta=(6)/(a^(2)-b^(2))and sin theta=(-8)/(a^(2)-b^(2))`
Squaring and adding, we get
`(64)/((a^(2)-b^(2))^(2))+(36)/((a^(2)-b^(2))^(2))=1`
`rArr 100=(a^(2)-b^(2))^(2)`
`rArra^(2)-b^(2)=10" "( :. agtb)`
`rArra^(2)e^(2)=10`
Hence, foci are `(+-sqrt(10),0)`
Promotional Banner

Similar Questions

Explore conceptually related problems

if the line x- 2y = 12 is tangent to the ellipse (x^(2))/(a^(2))+(y^(2))/(b^(2))=1 at the point (3,(-9)/(2)) then the length of the latusrectum of the ellipse is

If a normal to the ellipse (x^(2))/(a^(2))+(y^(2))/(b^(2))=1 is 4-3y=7 and its ecentricity is (sqrt(7))/(4) , then the volume of L.R can be

Find the equation of the normal to the ellipse (x^2)/(a^2)+(y^2)/(b^2)=1 at the positive end of the latus rectum.

Find the normal to the ellipse (x^2)/(18)+(y^2)/8=1 at point (3, 2).

If the area of the ellipse ((x^2)/(a^2))+((y^2)/(b^2))=1 is 4pi , then find the maximum area of rectangle inscribed in the ellipse.

Find the maximum area of the ellipse (x^2)/(a^2)+(y^2)/(b^2)=1 which touches the line y=3x+2.

If the line y=mx+c is a tangent to the ellipse x^(2)+2y^(2)=4 , find c

If normal at any point 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

Show that the line x-y + 4 =0 is a tangents to the ellipse x^(2) + 3y^(2) =12 . Also find the coordinates of the points of contact.

If any tangent to the ellipse (x^2)/(a^2)+(y^2)/(b^2)=1 intercepts equal lengths l on the axes, then find ldot