Find the equation of common tangent of `y^(2)=4axandx^(2)=4by`.

Options

A. `a^("1/3")x+b^("1/3")y=(ab)^("1/3")`
B. `a^("1/3")x+b^("1/3")y=(ab)^("1/3")=0`
C. `a^("2/3")x+b^("2/3")y=(ab)^("2/3")`
D. `a^("2/3")x+b^("2/3")y=(ab)^("2/3")=0`

Answer

B

Explanation

The equation of any tangent to `y^(2)=4ax` is `y=mx+a/m" …(i)"`
If it is a tangent to the parabola `x^(2)=4by`, then the equation
`x^(2)=4b(mx+a/m)" must have equal roots".`
`"or, "x^(2)-4bm x-(4ab)/m=0" must have equal roots".`
`rArr" "16b^(2)m^(2)+(16ab)/m=0" [On equating Disc. to zero]"`
`rArr" "m^(2)=-a/brArrm=-(a/b)^("1/3")`
Putting the value of m in (i), we get
`y=-(a/b)^("1/3")x-a^("2/3")b^("1/3")rArra^("1/3")x+b^("1/3")y+a^("1/3")b^("2/3")=0`
This is the equation of the common tangent.
ALITER The equation of any tangent to `y^(2)=4ax` is
`y=mx+a/m" ...(i)"`
`rArr" "x=y/m-a/m^(2)rArrx=m'y+(-a)m^(2),` where m' `=1/m`
If it touches the parabola `x^(2)=4by`, then we must have
`-am^(2)=b/(m')`
`rArr" "m^(3)=-b/arArr1/m^(3)=-b/arArrm^(3)=-a/brArrm=(a/b)^("1/3")`
Substituting the value of m in (i), we obtain tha equation of the required common tangent as
`a^("1/3")x+b^("1/3")y+b^("2/3")b^("2/3")=0`