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Question

Statement-1 : An equation of a common tangent to the parabola y2=163x and the ellipse 2x2+y2=4 is y=2x+23.

Statement-2 : If the line y=mx+43m, (m0) is a common tangent to the parabola y2=163x and the ellipse 2x2+y2=4, then m satisfies m4+2m2=24.

A
Statement-1 is false, Statement-2 is true.
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B
Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for statement-1.
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C
Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for statement-1.
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D
Statement-1 is true, Statement-2 is false.
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Solution

The correct option is B Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for statement-1.
The equation of ellipse is,
2x2+y2=4
2x24+y24=1
x22+y24=1
comparing with standard form,
x2a2+y2b2=1
we get,
a2=2 and b2=4
and equation of tangent to ellipse x2a2+y2b2=1 is,
y=mx±a2m2+b2
y=mx±2m2+4 ...(1)
also given that,
equation of tangent to the parabola y2=163x is,
y=mx+43m ...(2)
on comparing (1) and (2) we get,
43m=2m2+4
43=m2m2+4
squaring on both the sides,
48=2m4+4m2
2m4+4m248=0
m4+2m224=0
m4+2m2=24 this equation satisfies statement-2

On solving this equation
m4+2m224=0
m4+6m24m224=0
m2(m2+6)4(m2+6)=0
we get,
m2+6=0 and m24=0
m24=0 m=±2
the equation of common tangent are y=±2x±23
Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for statement-1.

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