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find the common tangent to a circle x^2+y^2=2a^2 and the parabola y^2=8ax

find the common tangent to a circle x^2+y^2=2a^2 and the parabola y^2=8ax

Grade:12th pass

2 Answers

Arun Kumar IIT Delhi
askIITians Faculty 256 Points
9 years ago
Hello Student,

Equation of tangent to the parabola in slope form:
y = mx + \frac{2a}{m}…..............(1)
Equation of tangent to the circle in slope form:
y = mx + \sqrt{2}a\sqrt{1+m^{2}}…................(2)
Equation (1), (2) must be same
\\\frac{2a}{m} = \sqrt{2}a\sqrt{1+m^{2}}\\ m^{4}+m^{2}-2=0\\ (m^{2}+2)(m^{2}-1)=0\\ m = \pm 1
Equation of common tangents:
y = x+2a
y = -x-2a
Thanks & Regards
Arun Kumar
Btech, IIT Delhi
Askiitians Faculty
Yash Chourasiya
askIITians Faculty 256 Points
3 years ago
Dear Studnet

Let common tangent to the curves be
643-2175_t.PNG
Now Distance from (0, 0) to the tangent line = Radius of circle​

643-1958_t1.PNG

I hope this answer will help you.
Thanks & Regards
Yash Chourasiya

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