Question

# If two tangents drawn from the point P to the parabola y2=12x be such that the slope of the tangent is double the other, then â€²Pâ€² lies on the curve

A
2y2=9x
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B
y2=9x
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C
2y2=27x
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D
y2=15x
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Solution

## The correct option is C 2y2=27xGiven a parabola ,y2=12xThere is a point P outside the parabola with coordinates(h,k).Now the equation of a tangent to the parabola y2=4ax is given byy=mx+am;m is the slope of tangentHere a=3y=mx+3mis the equation of any tangent to the parabola y2=12x.This tangent passes through point (h,k) hence⇒m2h−km+3=0−(i)So we can see that there are two possible values of ′m′ and hence we can have two tangents drawn to the parabola y2=12x from the point (h,k).Also the slope of one is twice that of the other.So say m1 & m2 are the roots of the quadratic equation(i)Then we have m1=2m2−(ii)m1+m2=kh−(iii)m1.m2=3h−(iv)Now, 2m22=3h−(v)3m2=kh⇒m2=k3h−(vi)Using (vi) we can reduce (v) as 2k29h2=3h⇒2k2=27hReplace k with y & h with x we have 2y2=27x

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