please answer this question as soon as possible Q A circle is drawn such that it passes - Maths - Conic Sections

Question

please answer this question as soon as possible

Q A circle is drawn such that it passes through the point 0, 1 and
touches the parabola y = 2x2 at (1, 2) tangents are
drawn from
the points p (8/3, 10/3) to touch the circle at a and b The
diameter of the circle circumscribing the triangle pab is ?

Shivam Mishra · Accepted Answer

Dear Student,
Please find below the solution to the asked query:

Let equation of cirlce be x2+y2+2gx+2fy+c=0Circle passes through 0,1, hence0+1+0+2f+c=0⇒2f+c=-1
iAs circle touches parabola at 1,2, hence 1,2 lies on circle⇒1+4+2g+4f+c=0⇒2g+4f+c=-5
iiAs circle and parabola will have common tangent at point of contact
Slope of tangent ofparabola at 1,2=Slope of tangent of circle at 1,2y=2x2Differentiatedydx=4x⇒dydx1,2=4×1=4x2+y2+2gx+2fy+c=0Differentiate2x+2y
dydx+2g+2f dydx=02x+2y dydx1,2+2g+2f
dydx1,2=0⇒2×1+2×2×4+2g+2f×4=0⇒2g+8f=-18⇒g+4f=-9
iiiSolving i,ii,iii, we can get value of g and f
Centre of cirlce is C-g,-fAs we already know that P83,103, A,B,C are concyclic, and ∠PAC=∠PBC=90°, hence P will be diameter which you caneasily find using C-g,-f and P83,103

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please answer this question as soon as possible Q. A circle is drawn such that it passes through the point 0, 1 and touches the parabola y = 2x2 at (1, 2) tangents are drawn from the points p (8/3, 10/3) to touch the circle at a and b. The diameter of the circle circumscribing the triangle pab is ?