wiz-icon
MyQuestionIcon
MyQuestionIcon
1
You visited us 1 times! Enjoying our articles? Unlock Full Access!
Question

The equation of tangent with slope m to the ellipse x2a2+y2b2=1 is given by y=mxa2+b2m2.


A

True

No worries! We‘ve got your back. Try BYJU‘S free classes today!
B

False

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses
Open in App
Solution

The correct option is B

False


Any line with slope m can be written y = mx +c. If we substitute y = mx + c in the equation x2a2+y2b2=1 we will get a quadratic. Since y = mx + c is a tangent, the quadratic should have only one solution, because tangent and the ellipse will intersect at only one point.

x2a2+(mx+c)2b2=1 has one solution or one root.

We will simplify this quadratic and say discriminant = 0

b2x2 + a2(mx+c)2 = a2b2

b2x2 + a2(m2x2+2mcx+c2)2 = a2b2

(b2 + a2m2) x2+ 2mca2x+a2c2 a2b2=0

= 0 (2mca2)24(b2+a2m2)a2(c2b2)=0

4m2c2a4 4a2(b2 +a2m2)(c2 b2) = 0

m2c2a2 b2c2a2m2c2+b4+b2a2m2) = 0

b2c2 + b4 + b2a2m2 = 0

c2 + b2 + a2m2 = 0

c2 = b2 + a2m2

or c =b2+a2m2


flag
Suggest Corrections
thumbs-up
0
Join BYJU'S Learning Program
similar_icon
Related Videos
thumbnail
lock
MATHEMATICS
Watch in App
Join BYJU'S Learning Program
CrossIcon