locus question (1 Viewer)

ek102 · Aug 16, 2016

If P(x,y) is any point on the line y=4x+3 show that the midpoint 'M' of OP has coordinates (x/2,1/2(4x+3)). Hence find the locus of M. (O is the origin), dont understand when it asks for the 'locus of M'.

InteGrand · Aug 16, 2016

ek102 said:
If P(x,y) is any point on the line y=4x+3 show that the midpoint 'M' of OP has coordinates (x/2,1/2(4x+3)). Hence find the locus of M. (O is the origin), dont understand when it asks for the 'locus of M'.

$\noindent It's asking for how the point $M$ moves as $P$ moves. Since we have parametric equations for $x_M$ and $y_M$, namely $x_M = \frac{t}{2}$, $y_M = \frac{1}{2} \left(4t+3\right)$, where $t \in \mathbb{R}$, we can eliminate this parameter to get a Cartesian equation for the locus.$

ek102 · Aug 16, 2016

im sorry i didnt get that last bit, would you please elaborate?

InteGrand · Aug 16, 2016

ek102 said:
im sorry i didnt get that last bit, would you please elaborate?

$\noindent We have the parametric equations for $M(x,y)$ as $x = \frac{t}{2}$, $y = \frac{1}{2}(4t+3)$, for $t\in \mathbb{R}$. From the $x$ equation, we have $t = 2x$. Substituting into the $y$ equation, we find$

$\begin{align*}y &= \frac{1}{2}\left(4\times 2x + 3\right)\\ &= 4x + \frac{3}{2}, \end{align*}$

$\noindent which implies that the locus of $M$ is the line with Cartesian equation $y = 4x +\frac{3}{2}$.$

locus question (1 Viewer)

ek102

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InteGrand

Well-Known Member

ek102

New Member

InteGrand

Well-Known Member

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