CameraIcon
CameraIcon
SearchIcon
MyQuestionIcon
MyQuestionIcon
1
You visited us 1 times! Enjoying our articles? Unlock Full Access!
Question

Find the condition for the line xcosα+ysinα=p to be a tangent to the ellipse x2a2+y2b2=1


Open in App
Solution

Step 1: Convert the line equation into slope-intercept form

Given line equation isxcosα+ysinα=p and ellipse is x2a2+y2b2=1.

The slope-intercept form of a line is given as,

y=mx+c

where, m is the slope and c is the intercept.

So,

xcosα+ysinα=pysinα=-xcosα+py=-xcosαsinα+psinα

On comparing it with y=mx+c we get,

m=-cosαsinα and c=psinα

Step 2: Evaluate the tangent condition

We know that,

If y=mx+c is a tangent to the ellipse x2a2+y2b2=1, then c2=a2m2+b2. Thus,

psinα2=a2-cosαsinα2+b2p2sin2α=a2cos2αsin2α+b2p2=a2cos2α+b2sin2α

Hence, the condition for the line xcosα+ysinα=p to be a tangent to the ellipse x2a2+y2b2=1 is that p2=a2cos2α+b2sin2α.


flag
Suggest Corrections
thumbs-up
27
Join BYJU'S Learning Program
similar_icon
Related Videos
thumbnail
lock
Line and Ellipse
MATHEMATICS
Watch in App
Join BYJU'S Learning Program
CrossIcon