Applications of Derivatives IITJEE Maths tutorial: practice problems part IV

Question 1.

If the point on y = x \tan {\alpha} - \frac{ax^{2}}{2u^{2}\cos^{2}{\alpha}}, where \alpha>0, where the tangent is parallel to y=x has an ordinate \frac{u^{2}}{4a}, then what is the value of \alpha?

Question 2:

Prove that the segment of the tangent to the curve y=c/x, which is contained between the coordinate axes is bisected at the point of tangency.

Question 3:

Find all the tangents to the curve y = \cos{(x+y)} for -\pi \leq x \leq \pi that are parallel to the line x+2y=0.

Question 4:

Prove that the curves y=f(x), where f(x)>0, and y=f(x)\sin{x}, where f(x) is a differentiable function have common tangents at common points.

Question 5:

Find the condition that the lines x \cos{\alpha} + y \sin{\alpha} = p may touch the curve (\frac{x}{a})^{m} + (\frac{y}{b})^{m}=1.

Question 6:

Find the equation of a straight line which is tangent to one point and normal to the point on the curve y=8t^{3}-1, and x=4t^{2}+3.

Question 7:

Three normals are drawn from the point (c,0) to the curve y^{2}=x. Show that c must be greater than 1/2. One normal is always the x-axis. Find c for which the two other normals are perpendicular to each other.

Question 8:

If p_{1} and p_{2} are lengths of the perpendiculars from origin on the tangent and normal to the curve x^{2/3} + y^{2/3}=a^{2/3} respectively, prove that 4p_{1}^{2} + p_{2}^{2}=a^{2}.

Question 9:

Show that the curve x=1-3t^{2}, and y=t-3t^{3} is symmetrical about x-axis and has no real points for x>1. If the tangent at the point t is inclined at an angle \psi to OX, prove that 3t= \tan {\psi} +\sec {\psi}. If the tangent at P(-2,2) meets the curve again at Q, prove that the tangents at P and Q are at right angles.

Question 10:

Find the condition that the curves ax^{2}+by^{2}=1 and a^{'}x^{2} + b^{'}y^{2}=1 intersect orthogonality and hence show that the curves \frac{x^{2}}{(a^{2}+b_{1})} + \frac{y^{2}}{(b^{2}+b_{1})} = 1 and \frac{x^{2}}{a^{2}+b_{2}} + \frac{y^{2}}{(b^{2}+b_{2})} =1 also intersect orthogonally.

More later,

Nalin Pithwa.

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