Question Video: Using the Angle between Two Straight Lines to Solve a Problem | Nagwa Question Video: Using the Angle between Two Straight Lines to Solve a Problem | Nagwa

# Question Video: Using the Angle between Two Straight Lines to Solve a Problem Mathematics • First Year of Secondary School

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If the acute angle between the straight lines whose equations are ππ¦ β 2π₯ + 19 = 0 and 9π₯ β 7π¦ β 8 = 0 is π/4, find all the possible values of π.

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### Video Transcript

If the acute angle between the straight lines whose equations are ππ¦ minus two π₯ plus 19 is equal to zero and nine π₯ minus seven π¦ minus eight is equal to zero is π by four, find all the possible values of π.

In this question, weβre given the equations of two straight lines, both given in general form. And we can see one of these straight lines has an unknown value of π. We need to determine this value of π. And to do this, weβre told that the acute angle between the two straight lines is π by four. And itβs worth noting we need to determine all of the possible values of π, so there may be multiple correct solutions or no solutions.

To do this, letβs start by recalling how we find the acute angle between two given straight lines. We know if πΌ is the acute angle between two straight lines with slopes π sub one and π sub two, then the tan of πΌ will be equal to the absolute value of π sub one minus π sub two divided by one plus π sub one times π sub two. If we know the slopes of the two lines π sub one and π sub two, then we can solve for πΌ by taking the inverse tangent of both sides of the equation. However, in this case, weβre not trying to find the value of πΌ. Weβre already told that πΌ is π by four. Instead, we need to find the value of π. Since we know the value of πΌ, we could substitute this into our equation. We can then use the two given equations to find expressions for the slopes.

Letβs start with the slope of the first line. We want to find the slope of the line ππ¦ minus two π₯ plus 19 is equal to zero. And we can note this line is given in general form, and itβs difficult to determine the slope of a line given in general form. So instead, letβs rearrange this into slopeβintercept form. We start by adding two π₯ to both sides of the equation and subtracting 19 from both sides of the equation. We get ππ¦ is equal to two π₯ minus 19.

Now, weβre going to divide through by π. This gives us that π¦ is equal to two π₯ over π minus 19 over π. And itβs worth noting weβre making an assumption here. Weβre assuming our value of π is nonzero. If our value of π is equal to zero, then this equation simplifies to give us negative two π₯ plus 19 is equal to zero. We can rearrange this to get π₯ is equal to 19 over two. In other words, this line would be a vertical line at 19 over two.

Since vertical lines do not have a defined slope, we canβt use our formula to determine the angle between two straight lines if one of the lines is vertical. So weβre going to need to treat this case separately. We can, however, determine the slope of the second line in the same way. We start by adding seven π¦ to both sides of the equation. We get seven π¦ is equal to nine π₯ minus eight. We then divide the equation through by seven. π¦ is equal to nine over seven π₯ minus eight over seven.

Before we use our formula, letβs now consider the case where π is equal to zero. Weβll start by sketching our first line. Itβs a vertical line at 19 over two. We know the angle between the two straight lines is π by four; weβre told this in the question. And we can notice that this vertical line is parallel to the π¦-axis. So the angle between this line and any other line will be the same as the angle between that line and the π¦-axis since itβs a transversal of parallel lines. And in particular, weβre told that this angle is π by four. So we can also see that the angle it makes with the positive π₯-axis will also be π by four. And the only line which makes an angle of π by four in this manner with the positive π₯-axis will have slope one.

It is worth noting here weβve made one small assumption. Weβve assumed that our line is positive slope. We get a very similar story if we assumed our line had negative slope. Instead, we would have found that the angle with the negative π₯-axis was π by four. So it would have had slope negative one. And in either case, we can show this is not whatβs happening in this case. Since weβve already found the slope of the second line, the slope of the second line in slopeβintercept form is its coefficient of π₯, nine over seven. And this is not negative one or positive one. So the value of π cannot be equal to zero. So weβve shown that π is nonzero, so letβs clear some space and carry on with this method.

We can find the slope of the two lines as the coefficients of π₯. π sub one is two over π and π sub two is nine over seven. We can now substitute these into our formula along with the fact that πΌ is equal to π by four. Substituting these in, we get the tan of π by four is equal to the absolute value of two over π minus nine over seven divided by one plus two over π times nine over seven. This is now an equation entirely in terms of π. So we can try and solve this for π.

So letβs start by simplifying the right-hand side of this equation. Letβs start with the numerator. We want to multiply two over π by seven over seven and nine over seven by π divided by π. Doing this and simplifying, we get 14 minus nine π all divided by seven π. We can also simplify the denominator. Two over π multiplied by nine over seven is equal to 18 divided by seven π. Therefore, the right-hand side of this equation simplifies to give us the absolute value of 14 minus nine π over seven π divided by one plus 18 over seven π. We can evaluate the left-hand side of the equation. The tan of π by four is just equal to one.

We still need to simplify the right-hand side of this equation. Weβll do this by multiplying both the numerator and denominator by seven π. Doing this and simplifying, we get that one is equal to the absolute value of 14 minus nine π divided by seven π plus 18. We can now solve this as an absolute value equation. We recall if the absolute value of π₯ is equal to one, then either π₯ is equal to one or π₯ is equal to negative one. This gives us two equations. Either 14 minus nine π over seven π plus 18 is equal to one or 14 minus nine π over seven π plus 18 is equal to negative one.

We can solve both of these equations for π. In the first equation, we multiply through by seven π plus 18. This gives us seven π plus 18 is equal to 14 minus nine π. Now, we can add nine π to both sides of the equation and subtract 18 from both sides of the equation. This gives us that 16π is equal to negative four. Finally, weβll divide the equation through by 16. This gives us that π is equal to negative four over 16, which simplifies to negative one-quarter. We can do the same in the second case. We multiply through by seven π plus 18. This gives us that negative seven π minus 18 is equal to 14 minus nine π.

We can now solve this equation by adding nine π to both sides of the equation and adding 18 to both sides of the equation. This gives us that two π is equal to 32. And we can solve for π by dividing both sides of the equation through by two. We get that π is equal to 16, and this gives us our final answer. If the acute angle between the two given straight lines is π by four, then there are only two possible values of π. Either π is negative one-quarter or π is equal to 16.

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