MediumMathematicsHigh School (11–12)Coordinate GeometryNumericalConceptual

Find the two bisectors of the angles between the lines 4x + 3y + 7 = 0 and -5x + 12y + 7 = 0…

Try it on paper first. A hint, the key idea and the worked solution are here when you need them.

Find the two bisectors of the angles between the lines $4x + 3y + 7 = 0$ and $-5x + 12y + 7 = 0$ . Decide which bisector goes through the angle that holds the origin, and which one bisects the acute angle between the lines.

Need a reminder? The key ideaShow

The bisectors of two lines are $\dfrac{a_1x + b_1y + c_1}{\sqrt{a_1^2 + b_1^2}} = \pm\dfrac{a_2x + b_2y + c_2}{\sqrt{a_2^2 + b_2^2}}$ . With $c_1, c_2 > 0$ , the plus sign goes through the angle that holds the origin, which is acute when $a_1a_2 + b_1b_2 < 0$ . The image of $(x_1, y_1)$ in $ax + by + c = 0$ is $(x_1 - 2at,\ y_1 - 2bt)$ with $t = \dfrac{ax_1 + by_1 + c}{a^2 + b^2}$ . A ray from $A$ to $B$ by way of a mirror strikes where the line from $A$ to the image of $B$ meets the mirror.

The worked solution

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Work it out on paper first. It is fine to get stuck or make a mistake: that is how we learn. When you are ready, go through the solution one step at a time, and check each step against your own.

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