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Question

# In the parabola y2=4ax, the tangent at the point P, whose abscissa is equal to the latus rectum meets the axis in T and the normal at P cuts the parabola again in Q then PT:PQ=3:5.

A
True
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B
False
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Solution

## The correct option is B FalseEquation of the parabola is y2=4ax(given)Latus rectum=4aLet P(at2,2at) be the coordinates of tangent.Given: Abscissa of the tangent at P= latus rectum meeting its axis at T⇒at2=4a⇒t2=4⇒t=±2Take t=2 then coordinates of tangent are P(4a,4a)Tangent at the point P is 2y=x+4a meets the axis in T⇒y=0⇒x+4a=0⇒x=−4a∴T is (−4a,0)Normal at P cuts at Q(at12,2at1) where t1=−t−−2t=−2−−22=−2−1=−3∴Q is (9a,−6a)Now,(PQ)2=(−6a−0)2+(9a−4a)2=36a2+169a2=125a2(PT)2=(4a+4a)2+(4a−0)2=64a2+16a2=80a2Ratio=(PT)2(PQ)2=80a2125a2=1625⇒PTPQ=45∴PT:PQ=4:5Hence the given statement is false.

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