Answer:
Option b - [tex]x^2-y^2=3[/tex]
Step-by-step explanation:
Given : The slope of the tangent line to a curve at any point (x, y) on the curve is x divided by y.
To find : What is the equation of the curve if (2, 1) is a point on the curve?
Solution :
We know that,
The slope of the tangent line is the derivative.
So, [tex]\frac{dy}{dx}=\frac{x}{y}[/tex]
or [tex]yy'=x[/tex]
Subtract yy' on both side of the equation,
[tex]x-yy'=0[/tex]
Now, Integrating both side of the equation but keeping concept of implicit differentiation,
[tex](\frac{1}{2})x^2-(\frac{1}{2})y^2=c[/tex]
Where, c is the constant.
Solving the equation,
[tex](\frac{1}{2})(x^2-y^2)=c[/tex]
Substitute x=2 and y=1 to find c,
[tex](\frac{1}{2})(2^2-1^2)=c[/tex]
[tex](\frac{1}{2})(3)=c[/tex]
[tex]c=\frac{3}{2}[/tex]
Substitute c back in the equation,
[tex](\frac{1}{2})(x^2-y^2)=\frac{3}{2}[/tex]
Solving,
[tex]x^2-y^2=3[/tex]
Therefore, The required equation is [tex]x^2-y^2=3[/tex]
So, Option b is correct.
