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Conic Section

Question
CBSEENMA11015591
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The normal to a 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

  • parabola

  • circle

  • hyperbola

Solution

D.

hyperbola

Equation space of space normal space is space straight Y minus straight y space equals space minus space dx over dy left parenthesis straight X minus straight x right parenthesis rightwards double arrow space straight G space identical to open parentheses straight x space plus space straight y dy over dx comma 0 close parentheses open vertical bar straight x plus straight y dy over dx close vertical bar space equals space vertical line 2 straight x vertical line rightwards double arrow straight y dy over dx space equals space straight x space or space straight y space dy over dx space equals space minus space 3 straight x straight y space dy space equals space straight x space dx space or space ydy space equals space minus space 3 xdx straight y squared over 2 space equals space straight x squared over 2 space plus space straight c space or space straight y squared over 2 space equals space minus fraction numerator 3 straight x squared over denominator 2 end fraction plus straight c straight x squared minus straight y squared space equals space minus 2 straight c 3 straight x squared space plus space straight y squared space equals 2 straight c

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