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For what values of `a` will the tangents to the parabola y^2=4ax from a point, not on the y axis ,will be normal to x^2=4y

For what values of `a` will the tangents to the parabola y^2=4ax from a point, not on the y axis ,will be normal to x^2=4y

Grade:12th pass

1 Answers

Vikas TU
14149 Points
7 years ago
Differntiate and put y also,
Β y^2=4ax
ydy/dx = 2a
root(ax)dy/dx = a
dy/dx = a/root(ax) = > root(a/x).....................(1)
Β 
Differntiate same as
x^2 = 4ay
2x =4dy/dx
x = 2dy/dx
dy/dx = x/2
normal will be:
x/2 * n = -1
n = -2/x …...............(2)
Β 
eqn. (1) Β and eqn. (2) are equal.
Β 
therfore,
Β 
-2/x = root(a/x)
squaring both sides,
4/x^2 = a/x
4/x^2 – a/x = 0
(4 – ax)/x^2 = 0
x cannot be zero
or
4 – ax = 0
ax = 4
x = 4/a
y = 4/a^2
put these x and y in y^2 = 4ax
u will get,
a^4 = 1
(a^2 + 1) (a^2-1) = 0
a^2 = 1
a = 1 Β and a = -1

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