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Question

# Let the line y=mx and the ellipse 2x2+y2=1 intersect at point P in the first quadrant. If the normal to this ellipse at P meets the co-ordinate axes at (−13√2,0) and (0,β), then β is equal to:

A
23
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B
23
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C
223
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D
23
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Solution

## The correct option is D √23 Let P≡(x1,y1) Given equation of ellipse is 2x2+y2=1 ⇒4x+2ydydx=0 ⇒dydx∣∣∣(x1,y1)=−2x1y1 Therefore, slope of normal at P(x1,y1) is y12x1 Equation of normal at P(x1,y1) is y−y1=y12x1(x−x1) It passes through (−13√2,0) ⇒−y1=y12x1(−13√2−x1) ⇒x1=13√2 ⇒y1=2√23 as P lies in first quadrant. Since (0,β) lies on the normal of the ellipse at point P, hence we get β−y1=−y12 ⇒β=y12=√23

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