Question
Question: The equation of normal at \[\left( at,\dfrac{a}{t} \right)\] to the hyperbola \[xy={{a}^{2}}\] is ...
The equation of normal at (at,ta) to the hyperbola xy=a2 is
(a) xt3−yt+at4−a=0
(b) xt3−yt−at4+a=0
(c) xt3+yt+at4−a=0
(d) xt3+yt−at4−a=0
Solution
For solving this question you should know about the equation of a normal. When the equation of curve is given by y=f(x), then finding the equation of normal line at a curve point A(x1,y1) starts from finding the slope. The slope of the normal to the curve at a point A(x1,y1) is A=A=(dxdy)y=y1,x=x1−1.
Complete step-by-step solution:
According to the question it is asked to us to finding the equation of normal at a point of the hyperbola and both are given as (at,ta) and xy=a2 respectively.
So, for determining the equation of normal:
Step 1: - Determine the equation of the curve y=f(x), once that is established, find (dxdy) from the given equation of the curve; y=f(x).
Step 2: - The next step is very important at it involves calculating the slope for the equation of the normal line to the curve at a point A(x1,y1). The formula for the slope of the normal is
m=(dxdy)y=y1,x=x1−1
Step 3: - The last step is replacing the value of slope in the equation (y−y1)=m(x−x1). After solving this you have successfully found the equation for the normal.
Consider, xy=a2
Differentiating both sides w.r.t x we get,