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The x–intercept of the tangent at any arbitrary point of the curve is proportional to
The line = 1 touches the curve y = be^{–x/a} at the point
If the subnormal at any point on y = a1 – n x^{n} is of constant length, then the value of n is
If the tangent at P of the curve y^{2} = x^{3} intersects the curve again at Q and the straight lines OP, OQ make angles a, b with the x–axis, where `O' is the origin, then tan a/tan b has the value equal to
The length of the normal to the curve x = a(q + sin q), y = a (1 – cos q), at q = is
The beds of two rivers (within a certain region) are a parabola y = x^{2} and a straight line y = x – 2. These rivers are to be connected by a straight canal. The coordinates of the ends of the shortest canal can be
If the area of the triangle included between the axes and any tangent to the curve x^{n} y = a^{n} is constant, then n is equal to
At (0, 0), the curve y^{2} = x^{3} + x^{2}
For the curve x = t^{2} – 1, y = t^{2} – t, the tangent line is perpendicular to xaxis where
Given curve is x = t^{2 }− 1,y = t^{2 }− t
Derivating w.r.to t we get
d x/ dt =2t (1)
and dy / dt = 2t − 1 (2)
dividing (2) by (1) we get
dy / dx = (2t − 1) / 2t
Therefore, the slope of the tangent is (2t − 1) / 2t
Given that the tangent is perpendicular to xaxis. Therefore, tangent is parallel to yaxis.
We know that slope of yaxis is infinity and the slopes of the two parallel lines are equal.
Therefore, slope of the tangent is infinity.
Hence, (2t − 1) / 2t = 1/0
⟹ 2t = 0
⟹ t = 0
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