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the normal to the curve at P(x,y) meets the x axis at G. If the distance of G from the origin is twice the abscissa of P , then the curve is a) ellipse b) parabola c) ellipse or hyperbola

the normal to the curve at P(x,y) meets the x axis at G. If the distance of G from the origin is twice the abscissa of P , then the curve is
a) ellipse
b) parabola
c) ellipse or hyperbola

Grade:Upto college level

4 Answers

Ajay Verma
askIITians Faculty 33 Points
10 years ago
solution:

for any curve the equation of normal at ( X1, Y1)

(y- y1) = -1/m (x- X1) {where m = slope of tengent or say dy/dx at (x1, Y1) }

normal meets the X axis at G..
so
( 0 - y1) = -1/m (x- X1)
so x = m*Y1 + X1
so coordinates of G( m*Y1 + X1 , 0)

now ...distance of G from the origin is twice the abscissa of P
so
2X1 = m*Y1 + X1
X1 = m*Y1
m = X1/Y1
dy/dx (at X1, Y1) = X1/Y1

dy/dx = x/y
ydy = x dx
integrate both sides
y2/2 = x2/2 + c

x2/2 - y2/2 = c

so hyperbola..


Thanks and Regards,
Ajay verma,
askIITians faculty,
IIT HYDERABAD
vansh kharbanda
20 Points
10 years ago
thank you
vansh kharbanda
20 Points
10 years ago
but sir the answer says it can be both ellipse as well as hyperbola, so can u please prove this as an ellipse too.
Ayushi Agarwal
36 Points
6 years ago
The equation is |x1 + my1|=2|x1|So there are 2 equations possible one with positive and the other with negative

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