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प्रश्न
Find the equation of a curve passing through the point (0, -2) given that at any point (x, y) on the curve, the product of the slope of its tangent and y-coordinate of the point is equal to the x-coordinate of the point.
उत्तर
According to the question, y `dy/dx` = x (where `dy/dx` is the slope of the tangent.)
y dy = x dx
On integrating
⇒ `int y dy int x dx`
`y^2/2 = x^2/2 + C` .... (i)
∵ The curve passes through the point (0, -2)
∴ Putting x = 0, y = - 2,
`4/2 = 0 + C`
⇒ C = 2
On putting C = 2 in equation (i),
`y^2/2 = x^2/2 + 2`
`=> y^2 = x^2 + 4`
`=> y^2 - x^2 = 4`
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