Question 538·Hard·Algebra

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In the $xy$ -plane, line $m$ passes through the points $(0,6)$ and $(4,-2)$ . Line $n$ is perpendicular to line $m$ and intersects line $m$ at point $P$ . If point $P$ is 3 units from the origin and the $x$ -intercept of line $n$ is negative, what is the $x$ -intercept of line $n$ ?

A point a fixed distance from the origin lies on a circle, and a line can cross that circle twice. Find the given line’s equation, then graph it with the circle in Desmos to locate both possible points. Use the negative-intercept condition to choose the point before drawing the perpendicular line. That condition describes the intercept, not the point’s $x$ -coordinate.

Hints

Hint 1
A line’s slope is the change in $y$ divided by the change in $x$ . Find that change between the two given points; the point with $x=0$ then gives you the line’s $y$ -intercept.
Hint 2
A point $3$ units from the origin satisfies $x^2+y^2=3^2$ : its horizontal and vertical distances form the legs of a right triangle. Graph this circle with line $m$ to find both possible locations of $P$ .
Hint 3
If one possible $P$ lies on the $x$ -axis, every line through it has its x-intercept there. Check that intercept’s sign before choosing which point to use.

Step-by-step

Graph the possible points, then draw the perpendicular line
Step 1
Find the slope of line $m$
The slope is the change in $y$ divided by the change in $x$ . From $(0,6)$ to $(4,-2)$ , type $\frac{-2-6}{4-0}$ in Desmos. It gives $-2$ , so line $m$ falls $2$ units for every $1$ unit you move right.
Step 2
Write and graph line $m$
At $(0,6)$ , $x=0$ , so $6$ is the y-intercept, where the line meets the vertical axis. With slope $-2$ , type $y=-2x+6$ . Desmos graphs line $m$ through the two given points.
Step 3
Find both possible locations of $P$
Being $3$ units from the origin means a point $(x,y)$ lies on the circle $x^2+y^2=3^2=9$ . The squared horizontal and vertical distances add to the squared distance from the origin. Type $x^2+y^2=9$ beside line $m$ and click both intersections. Desmos shows $(3,0)$ and $(1.8,2.4)$ , the two possible locations of $P$ .
Step 4
Use the sign condition to choose $P$
If $P=(3,0)$ , line $n$ passes through that point on the $x$ -axis, so its $x$ -intercept is positive $3$ . That breaks the given condition. Use $P=(1.8,2.4)$ instead. Test the sign of the requested intercept, not the sign of $P$ ’s $x$ -coordinate.
Step 5
Find the perpendicular slope
Line $n$ is perpendicular to line $m$ , meaning they meet at a right angle. Perpendicular slopes multiply to $-1$ , so flip and change the sign of $-2$ : line $n$ has slope $\frac12$ . Keeping $-2$ would make the lines parallel instead.
Step 6
Graph line $n$ and read its $x$ -intercept
Use slope $\frac12$ and $P=(1.8,2.4)$ to type $y-2.4=\frac12(x-1.8)$ . This line passes through $P$ because both sides equal $0$ when $x=1.8$ and $y=2.4$ . Click where it meets the $x$ -axis: Desmos shows $(-3,0)$ . The $x$ -intercept of line $n$ is $-3$ . Choice B.

Question 538·Hard·Algebra

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Step-by-step

Lessons that teach this

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Strategy

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Step-by-step

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