THETA DIVISION

Question 86

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Question

The shortest distance between a point

(x_0,y_0)

and a line

Ax+By+C=0

is given by

\frac{|Ax_0+By_0+C|}{\sqrt{A^2+B^2}}.

Let

S

be the set of all points

P

such that the product of the distance from

P

to the line

y=x

and the distance from

P

to the line

y=-\frac34x

\frac1{5\sqrt2}

. Set

S

consists of two slightly rotated hyperbolas such that one of which has a transverse axis with a positive slope (call this hyperbola

H

). Compute the smallest positive

x

-coordinate of any point on

H

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Question

The shortest distance between a point

(x_0,y_0)

and a line

Ax+By+C=0

is given by

\frac{|Ax_0+By_0+C|}{\sqrt{A^2+B^2}}.

Let

S

be the set of all points

P

such that the product of the distance from

P

to the line

y=x

and the distance from

P

to the line

y=-\frac34x

\frac1{5\sqrt2}

. Set

S

consists of two slightly rotated hyperbolas such that one of which has a transverse axis with a positive slope (call this hyperbola

H

). Compute the smallest positive

x

-coordinate of any point on

H