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# Find The Equation And The Length Of The Common Tangents To Hyperbola

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## Question

### Solution

Correct option is

Similarly tangent at any point  on 2nd hyperbolas is

If (i) and (ii) are common tangents then they should be identical. Comparing the coefficients of x and y

Hence the points of contact are

Length of common tangent i.e., the distance between the above points is  and equation of common tangent on putting the values of  in (i) is

Alternative Method : The given two hyperbolas are

we know that

is tangent to (i) for all

will be tangent to (ii)

For common tangents to (i) and (ii) the lines (iii) and (iv) must be identical

∴ The equation of common tangent lines are

equation of tangent to (i) at (x1y1) is

Comparing (v) and (vi), then

and equation of tangent to (ii) at (x2y2) is

Comparing (v) and (vii), then

Hence the points of contact are

Hence the length of common tangent is

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#### SIMILAR QUESTIONS

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