Question

# A graph of ${\text{r = 10}cos\theta }$, where r and $\theta$ are the polar coordinates is, A) A circleB) An ellipseC) A horizontal lineD) A hyperbolaE) A vertical line

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Hint: Convert the polar coordinates into Cartesian coordinates using ${\text{x = r}cos\theta }$ and ${\text{y = r}sin\theta }$. Use both the coordinates of x and y to convert ${\text{r = 10}cos\theta }$ into the terms of x and y and try to mould it into the equation of possible curve.

Complete step by step solution: As we know, from the above hint that ${\text{x = r}cos\theta }$ . Replace the value of from $x = rcos\theta$ into $r = 10cos\theta$
Using ${cos\theta = }\dfrac{{\text{x}}}{{\text{r}}}$ in the given polar coordinate equation,
And so on simplifying we can get ${{\text{r}}^{\text{2}}}{\text{ = 10x}}$
As we know that equation of circle is $\sqrt {{{\text{x}}^{\text{2}}}{\text{ + }}{{\text{y}}^{\text{2}}}} {\text{ = r}}$
Put the value of ${{\text{r}}^{\text{2}}}$in the equation of the circle . and so the equation will be simplified to
${{\text{x}}^{\text{2}}}{\text{ + }}{{\text{y}}^{\text{2}}}{\text{ = 10x}}$
Make the perfect square of both the variables as to convert it in the equation of circle if possible,
${{\text{x}}^{\text{2}}}{\text{ - 10x + }}{{\text{y}}^{\text{2}}}{\text{ + 25 - 25 = 0}}$
${\left( {{\text{x - 5}}} \right)^{\text{2}}}{\text{ + }}{{\text{y}}^{\text{2}}}{\text{ = 25}}$
So from the above equation we can state that through the above given polar coordinates if converted into Cartesian coordinates then it will be a circle.

Hence the above obtained is the equation of circle with radius 5 and centre${\text{}}\left( {{\text{5,0}}} \right){\text{.}}$

Note: use the conversion of polar coordinate into Cartesian coordinate properly. Using the conversion, replace the values of r and x from equations and mould it properly into the possible equation of the curve. Form the equation properly so that the equation of the curve can be easily obtained. Also remember the general expression of circle, ellipse, hyperbola etc.
In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The x and y coordinates of a point measure the respective distances from the point to a pair of perpendicular lines in the plane called the coordinate axes, which meet at the origin.