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Find the common tangent of x^(2) + y^(2...

Find the common tangent of `x^(2) + y^(2) = 2a^(2)` and `y^(2) = 8ax`.

Text Solution

Verified by Experts

Given equation are
`x^(2) + y^(2) = 2a^(2) " "…(i)`
and `y^(2) = 8ax" "…(ii)`
The line y = mx + 2a/m is a tangent to the given parabola (ii).
This will touch the circle (i), if the perpendicular upon it from (0, 0) the centre of the circle is equal to `sqrt(2)a`, the radius of the circle.
i.e., `(|(2a)/(m)|)/(sqrt(1+m^(2)))=sqrt(2)a or 4a^(2) = m^(2)(1+m^(2))2a^(2)`
or `m^(2) (m^(2) +1) -2 = 0 or m^(4) + m^(2) - 2 = 0`
or `(m^(2) -1) (m^(2) + 2) = 0`
`rArr m^(2) = 1 {therefore m^(2) + 2 ne 0}`
`therefore m = pm 1`
Hence the common tangents are `y = pm (x + 2a)`
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