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Chapter 15: Problem 1426

The curve represented by \(\mathrm{x}=3(\cos \mathrm{t}+\sin \mathrm{t})\); \(\mathrm{y}=4(\cos \mathrm{t}-\sin \mathrm{t})\) is (a) circle (b) parabola (c) ellipse (d) hyperbola

Short Answer

Expert verified
The curve represented by the given parametric equations is an ellipse, because after eliminating the parameter \(t\) and simplifying the expression, we obtain an equation of the form \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) with \(a^2 = 1\) and \(b^2 = \frac{9}{2}\). Therefore, the correct answer is (c) ellipse.

Step by step solution

01

Express \(\cos t\) and \(\sin t\) in terms of \(x\) and \(y\)

From the given parametric equations, we can express \(\cos t\) and \(\sin t\) in terms of \(x\) and \(y\): \[\cos t = \frac{x}{3} - \frac{y}{4}\] \[\sin t = \frac{x}{3} + \frac{y}{4}\]
02

Square and sum the two equations

Now, we square both equations and sum them to eliminate the parameter \(t\): \[\left(\cos t\right)^2 + \left(\sin t\right)^2 = \left(\frac{x}{3} - \frac{y}{4}\right)^2 + \left(\frac{x}{3} + \frac{y}{4}\right)^2\] Since \(\cos^2 t + \sin^2 t = 1\), we can simplify the above equation to obtain the type of curve represented by these parametric equations.
03

Simplify the resulting equation and determine the type of curve

\begin{align*} 1 &= \left(\frac{x}{3} - \frac{y}{4}\right)^2 + \left(\frac{x}{3} + \frac{y}{4}\right)^2 \\ &= \frac{x^2}{9} - \frac{xy}{6} + \frac{y^2}{16} + \frac{x^2}{9} + \frac{xy}{6} + \frac{y^2}{16}\\ &= \frac{2x^2}{9} + \frac{2y^2}{16} \end{align*} Divide both sides by 2, and then multiply by 9 and 16 to eliminate the fractions: \[9\left(\frac{x^2}{9}\right) + 16\left(\frac{y^2}{16}\right) = 9(1)\] \[\frac{x^2}{1} + \frac{y^2}{\frac{9}{2}} = 1\] This is an equation of an ellipse because it is in the form \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), with \(a^2 = 1\) and \(b^2 = \frac{9}{2}\). So, the correct answer is: (c) ellipse

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Most popular questions from this chapter

If two circle \((\mathrm{x}-1)^{2}+(\mathrm{y}-3)^{2}=\mathrm{a}^{2}\) and \(\mathrm{x}^{2}+\mathrm{y}^{2}-8 \mathrm{x}+2 \mathrm{y}+8=0\) intersect in two distinct points, then (a) \(2<\mathrm{a}<8\) (b) \(a>2\) (c) \(\mathrm{a}<2\) (d) \(\mathrm{a}=2\)

The equation of the tangent to the curve \(4 \mathrm{x}^{2}-9 \mathrm{y}^{2}=1\). Which is parallel to \(5 \mathrm{x}-4 \mathrm{y}+7=0\) is (a) \(30 \mathrm{x}-24 \mathrm{y}+17=0\) (b) \(24 \mathrm{x}-30 \mathrm{y} \pm \sqrt{(161)}=0\) (c) \(30 \mathrm{x}-24 \mathrm{y} \pm \sqrt{(161)}=0\) (d) \(24 \mathrm{x}+30 \mathrm{y} \pm \sqrt{(161)}=0\)

The locus of a point \(\mathrm{P}(\alpha, \beta)\) moving under the condition that the line \(\mathrm{y}=\alpha \mathrm{x}+\beta\) is a tangent to the hyperbola \(\left(x^{2} / a^{2}\right)-\left(y^{2} / b^{2}\right)=1\) is (a) a circle (b) a parabola (c) an ellipse (d) a hyperbola

The equation \(2 \mathrm{x}^{2}+3 \mathrm{y}^{2}-8 \mathrm{x}-18 \mathrm{y}+35=\mathrm{k}\) represents (a) parabola if \(\mathrm{k}>0\) (b) circle if \(\mathrm{k}>0\) (c) a point if \(\mathrm{k}=0\) (d) a hyperbola if \(\mathrm{k}>0\)

The point of intersection of the tangents at the ends of the latus rectum of the parabola \(\mathrm{y}^{2}=4 \mathrm{x}\) is ....... (a) \((-1,0)\) (b) \((1,0)\) (c) \((0,0)\) (d) \((0,1)\)
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