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Length of common tangents to the hyperbo...

Length of common tangents to the hyperbolas `x^2/a^2-y^2/b^2=1` and `y^2/a^2-x^2/b^2=1` is

A

`y=pm3xpmsqrt(b^2-a^2)`

B

`y=pmxpmsqrt(a^2-b^2)`

C

`y=pm2xpmsqrt((4a^2-b^2))`

D

`y=pmxpmsqrt((a^2+b^2))`

Text Solution

Verified by Experts

The correct Answer is:
B
Let y=mx+c be a common tangent of hyperbolas
`x^2/a^2-y^2/b^2=1`…(i) and `y^2/a^2-x^2/b^2=1` …(ii)
Condition of tangency for equation (i) is `c^2=a^2 m^2 -b^2` …(iii)
And condition of tangency for equation (ii) is `c^2=a^2-b^2m^2`…(iv)
From equation (iii) and (iv),
`a^2m^2 -b^2 =a^2-b^2m^2 rArr a^2(m^2-1)+b^2 (m^2-1)=0 rArr (a^2+b^2) (m^2-1)=0`
`because (a^2+b^2) ne 0 " " therefore m^2-1=0 rArr m= pm1`
From equation (iii), `c^2=a^2-b^2 " " therefore c=pmsqrt((a^2-b^2))`
Hence, equation of common tangents are `y=pm x pm sqrt((a^2-b^2))`
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