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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
8 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
8 years ago
thank you
vansh kharbanda
20 Points
8 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
4 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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