Question
Question: The equations of \[{L_1}\] and \[{L_2}\] are \[y = mx\] and \[y = nx\], respectively. Suppose \[{L_1...
The equations of L1 and L2 are y=mx and y=nx, respectively. Suppose L1 makes twice as large of an angle with the horizontal (measured counterclockwise from the positive –axis) as does L2 and that L1 has 4 times the slope of L2. If L1 is not horizontal, then the value of the product(mn) equals.
A. 22
B. −22
C. 2
D. −2
Solution
First, we will take m=tan2θ and n=tanθ and then use the property tan2θ=1−tan2θ2tanθ to simplify the values of m and n. Then substitute the obtained values in the product mn.
Complete step by step solution:
Given that L1 is y=mx and L2 is y=nx.
We know that the slope of L1 is m and L2 is from the above equations.
Since it is given that the L1 has 4 times the slope of L2, m=4n.
Take m=tan2θ and n=tanθ, we get
m=4n
We will use the property of tangential function, that is, tan2θ=1−tan2θ2tanθ, where θ is the angle.
Using the above property in the above equation m=tan2θ, we get
m=1−tan2θ2tanθ
Taking n=tanθ in the above equation, we get
m=1−n22n
Substituting this value of m in the given equation m=4n, we get
4n=1−n22n
Dividing the above equation by 2n on each of the sides, we get
⇒2=1−n21
Cross-multiplying the above equation, we get
⇒2(1−n2)=1 ⇒2−2n2=1Subtracting the above equation by 2 on both sides, we get
⇒2−2n2−2=1−2 ⇒−2−2n2=−2−1 ⇒n2=21Dividing the above equation by −2 on each of the sides, we get
Taking square root in the above equation on both sides, we get
⇒n=±21
Since we know that the slope of a line can never be negative, so the value −21 is discarded.
Substituting the positive value of n in the given equation m=4n, we get
m=24
Now we will find the product mn from these values of m and n.
mn=24×21 ⇒mn=24 ⇒mn=2Thus, the product mn equals 2.
Hence, the correct option is C.
Note:
In this question, we can also solve it by taking the angle made by y=mx with positive direction of –axis is tan−1m and the angle made by line by y=nx is tan−1n.
Now, we will take tan−1m=2tan−1n.
We will use the property of tangential function, that is, 2tan−1a=tan−11−a22a.
Using the above property in our assumed equation tan−1m=2tan−1n, we get
⇒tan−1m=tan−11−n22n ⇒m=1−n22nSubstituting this value of m in the equation m=4n, we get
⇒1−n22n=4n
Dividing the above equation by 2n on each of the sides, we get
⇒1−n21=2
Cross-multiplying the above equation, we get
⇒1=2(1−n2) ⇒1=2−2n2Subtracting the above equation by 2 on both sides, we get
⇒1−2=2−2n2−2 ⇒−1=−2n2Dividing the above equation by −2 on each of the sides, we get
⇒−2−1=−2−2n2 ⇒21=n2 ⇒n2=21Taking square root in the above equation on both sides, we get
⇒n=±21
Since we know that the slope of a line can never be negative, so the value −21 is discarded.
Substituting the positive value of n in the given equation m=4n, we get
m=24
Now we will find the product mn from these values of m and n.
mn=24×21 ⇒mn=24 ⇒mn=2Thus, the product mn equals 2.