To find the point on the curve
x2=xy which is closest to the point
(0,5), we need to minimize the distance between a point
(x,y) on the curve and
(0,5). We will use the distance formula.
The distance
d between the points
(x,y) and
(0,5) is given by:
d=√x2+(y−5)2 We want to minimize this distance. Since minimizing the distance is equivalent to minimizing the square of the distance, we can minimize:
d2=x2+(y−5)2 Using the given equation of the curve
x2=xy, we can express
y in terms of
x. Thus, solving for
y,
y=‌=x Substitute
y=x in the expression for
d2 :
d2=x2+(x−5)2Expanding and simplifying,
d2=x2+x2−10x+25=2x2−10x+25 To find the value of
x that minimizes
d2, we take the derivative of
d2 with respect to
x and set it to zero:
‌(2x2−10x+25)=4x−10=0 Solving for
x,
‌4x−10=0‌4x=10‌x=‌Given that
y=x, the corresponding
y value is:
y=‌ Hence, the point on the curve which is closest to
(0,5) is:
(‌,‌)Therefore, the correct option is:
Option A
(‌,‌)