a little tricky question
Calculus
Level
3
What are the equations of the mutual tangent lines to and ?
there is no mutual tangential
and
and
and
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1 solution
lets find the intersection points of the two linear equations x^2 -6x +10=4x- x^2-7 no real solutions the two curves don't intersect lets suppose that the tangent intersects the first curve at (A , B) and the second curve at (C, D) the slope of the tangent line would be (D-B)/(C-A)
and this is equal to (4C-C^2-7)-(A^2-6A+10) / (C-A) (1) the slope of the first tangent is equal to ∆y⁄∆x =(2A-6) (2) the slope of the second tangent is equal to ∆y⁄∆x =(4-2C) (3) solving (2) and (3) together so (A+C=5) (4) FROM (1) ,(2),(3) (4C-C^2-7)-(A^2-6A+10)/(C-A) =(2A-6) FROM (4) SO (20-4A-25+10A-A^2-7-A^2+6A-10)=(-4A^2+22A-30) (A^2-5A+4=0) so ((A-1)(A-4)=0) IF A=1 SO B =5 so the point is (1,5) so the slope =-4 so the equation of the tangent is 4x+y-9=0 OR A=4 SO B=2 SO THE POINT IS (4,2) SO THE SLOPE =2 so the equation of the tangent is 2x-y-6=0