Question

# If the x-intercept of the focal chord of the parabola y2−2y−4x+5=0 is 114, then find the length of chord.

A
152
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B
252
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C
254
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D
154
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Solution

## The correct option is C 254y2−2y−4x+5=0⇒y2−2y+1=4x−4⇒(y−1)2=4(x−1) Vertex :(1,1), Focus :(2,1) Focal chord passes through (114,0) Therefore, equation of focal chord is yx−114=12−114 ⇒4x+3y=11 Let a point on parabola, through which focal chord passes be (1+t2,1+2t) ∵4x+3y=11⇒4(1+t2)+3(1+2t)=11⇒(2t−1)(t+2)=0⇒t=−2,12 Therefore, the end points of the focal chord are : (1+t21,1+2t1),(1+t22,1+2t2) Length of chord, L=√(t22−t21)2+(2t2−2t1)2 ⇒L=√(4−14)2+(−4−1)2=254

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