91Ó°ÊÓ

The point on the curve \(y^{2}=x\), the tangent at which values an angle of \(45^{\circ}\) with \(x\)-axis will be given by (a) \(\left(\frac{1}{2}, \frac{1}{4}\right)\) (b) \(\left(\frac{1}{2}, \frac{1}{2}\right)\) (c) \((2,4)\) (d) \(\left(\frac{1}{4}, \frac{1}{2}\right)\)

Find the acute angles between the curves \(y=\left|x^{2}-1\right|\) and \(y=\left|x^{2}-3\right|\) at their points of intersection.

Find the equation of the tangent to the curve \(y=\frac{4}{x^{2}+2}\) at \(x=0 .\).

Find all the tangents to the curve \(y=\cos (x+y)\), \([-2 \pi, 2 \pi]\) that are parallel to the line \(x+2 y=0\).

If the tangent to the curve \(x+y=e^{x y}\) be parallel to the \(y\)-axis, then the point of contact is (a) \((1,0)\) (b) \((0,1)\) (c) \((1,1)\) (d) None

Find the equations of the common tangents of the circle \(x^{2}+y^{2}-6 y+4=0\) and the parabola \(y^{2}=x\) .

Find the equation of the normal to the curve \(x^{3}+y^{3}=6 x y\) at \((3,3)\).

If the parametric equation of curve is given by \(x=e^{t} \cos t, y=e^{t} \sin t\), then the tangent to the curve at the point \(t=\frac{\pi}{4}\) values with the axis of the angle is (a) 0 (b) \(\frac{\pi}{4}\) (d) \(\frac{\pi}{2}\) (c) \(\frac{\pi}{3}\)

The curve \(y-e^{x y}+x=0\) has a vertical tangent at the point (a) \((1,1)\) (b) at no point (c) \((0,1)\) (d) \((1,0)\)

If \(p_{1}\) and \(p_{2}\) be the lengths of perpendiculars from the origin on the tangent and normal respectively at any point \((x, y)\) on the curve \(x^{2 / 3}+y^{2 / 3}=a^{2 / 3}\), then show that \(4 p_{1}^{2}+p_{2}^{2}=a^{2}\).