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Hyperbola question

2015 · Shift 2 · Q30

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PYQ Library
JEE Advanced
Mathematics
Hyperbola
2015 · Shift 2 · Q30

Hyperbola question

2015 · Shift 2 · Q30

JEE AdvancedMathematicsHyperbolaMultiple correct+4 / −1

Consider the hyperbola

H:{x^2} - {y^2} = 1

and a circle

S

with center

N\left( {{x_2},0} \right)

. Suppose that

H

and

S

touch each other at a point

P\left( {{x_1},{y_1}} \right)

with

{{x_1} \gt 1}

and

{{y_1} \gt 0}

. The common tangent to

H

and

S

at

P

intersects the

x

-axis at point

M

. If

(l, m)

is the centroid of the triangle

PMN

, then the correct expressions(s) is(are)

A
${{dl} \over {d{x_1}}} = 1 - {1 \over {3x_1^2}}$ for ${x_1} \gt 1$
B
${{dm} \over {d{x_1}}} = {{{x_1}} \over {3\left( {\sqrt {x_1^2 - 1} } \right)}}$ for ${x_1} \gt 1$
C
${{dl} \over {d{x_1}}} = 1 + {1 \over {3x_1^2}}$ for ${x_1} \gt 1$
D
${{dm} \over {d{y_1}}} = {1 \over 3}$ for ${y_1} \gt 0$

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