Q. 243.5( 2 Votes )

A curve is such that the length of the perpendicular from the origin on the tangent at any point P of the curve is equal to the abscissa of P. Prove that the differential equation of the curve is and hence find the curve.

Answer :



It is a homogenous equation,


Putting y = kx



So,











Putting the value of k





Differentiating with respect to x,




Let (h,k) be the point where tangent passes through origin and the length is equal to h. So, equation of tangent at (h,k) is




2ky – 2k2 = cx – ch – 2hx + 2h2


x(c – 2h) – 2ky + 2k2 –hc + 2h2 = 0


x(c – 2h) – 2ky + 2(k2 –2h) – hc = 0


x(c – 2h) – 2ky + 2(ch) – hc = 0 ( h2 + k2 = ch as (h,k) on th curve)


x(c – 2h) – 2ky + hc = 0


Now, Length of perpendicular as tangent from origin is








Hence, x2 + y2 = cx is the required curve.


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