Courses
Courses for Kids
Free study material
Offline Centres
More
Store Icon
Store
seo-qna
SearchIcon
banner

If the equation to the circle, having double contact with the ellipse x2a2+y2b2=1 (having eccentricity e) at the needs of a latus rectum, is x2+y2mae3x=a2(1e2e4). Find m.

Answer
VerifiedVerified
494.7k+ views
1 likes
like imagedislike image
Hint: We will first calculate x1 and x2 from the equation of normal to ellipse to calculate the x – coordinate and then we will calculate the distance in which the radius of the circle will lie. After that, we will calculate the length of the radius of the circle using the distance formula then form the equation of the circle using the x – coordinate and the radius of the circle. We will compare both the equations of the circle (obtained and the given) for the value of m.

Complete step-by-step answer:
 Double contact refers to a point where both the curves, circle and ellipse, will have the same slope and hence same tangent and normal lines.
seo images

Normal to the circle will also be the normal to the curve therefore, equation of the normal to the ellipse will be:
a2xx1b2yy1=a2b2
At x1 = ae and y1 = b2a (x1 and y1 are the endpoints of the latus rectum)
This equation of normal to the ellipse will also be the normal to the circle.
And by symmetry, we can say that the centre of the circle will lie at the x – axis.
Putting the value of x1 and y1 in the equation of the normal, we get
a2xaeb2yb2a=a2b2axey=a2b2
The x – coordinate will be ae3. Therefore, the radius of the circle will lie between (ae,b2a) and (ae3, 0).
Now, we know that 1e2=b2a2
b2=a2(1e2)b4=a4(1e2)2
 Now, using the distance formula for calculating the radius of the circle, we get
r=(x2x1)2+(y2y1)2
r=(ae3ae)2+(0b2a)2r=(a2e6+a2e22a2e4)+(b4a2)
Putting the value of b4 in the above equation, we get
 r=(a2e6+a2e22a2e4)+(a4(1e2)2a2)r=(a2e6+a2e22a2e4)+(a2(1+e42e2))r=(a2e6+a2e22a2e4)+(a2+a2e42a2e2)r=a2e6+a2a2e4a2e2
Squaring both sides, we get
r2=a2e6+a2a2e4a2e2
Now, we know that the equation of the circle is (x – x1)2 + (y – y1)2 = r2
Substituting the values of x1, y1 and r, we get
(xae3)2+(y0)2=a2e6+a2a2e4a2e2x2+a2e62ae3x+y2=a2e6+a2a2e4a2e2x2+y22ae3x=a2a2e2a2e4x2+y22ae3x=a2(1e2+e4)
Comparing it with the given equation of the circle: x2+y2mae3x=a2(1e2e4)
We get the value of m = 2.

Note: In such questions, you may get confused at a lot of places like while reducing the x – coordinate and while calculating the radius from the distance formula using the coordinates. Be careful in simplification of the equation of radius.
Latest Vedantu courses for you
Grade 10 | CBSE | SCHOOL | English
Vedantu 10 CBSE Pro Course - (2025-26)
calendar iconAcademic year 2025-26
language iconENGLISH
book iconUnlimited access till final school exam
tick
School Full course for CBSE students
PhysicsPhysics
Social scienceSocial science
ChemistryChemistry
MathsMaths
BiologyBiology
EnglishEnglish
₹38,500 (9% Off)
₹35,000 per year
Select and buy