Point P partitions directed line segment AB in the ratio of 3 to 4. If A(-9, -9) and B(5,-2), find the coordinates of point P.
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Answer:
Our two intersection points are:
Step-by-step explanation:
We want to find where the two graphs given by the equations:
Intersect.
When they intersect, their <em>x-</em> and <em>y-</em>values are equivalent. So, we can solve one equation for <em>y</em> and substitute it into the other and solve for <em>x</em>.
Since the linear equation is easier to solve, solve it for <em>y: </em>
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Substitute this into the first equation:
Simplify:
Square. We can use the perfect square trinomial pattern:
Multiply both sides by 16:
Combine like terms:
Isolate the equation:
We can use the quadratic formula:
In this case, <em>a</em> = 25, <em>b</em> = -22, and <em>c</em> = -159. Substitute:
Evaluate:
Hence, our two solutions are:
We have our two <em>x-</em>coordinates.
To find the <em>y-</em>coordinates, we can simply substitute it into the linear equation and evaluate. Thus:
And:
Thus, our two intersection points are: