First slide

Pair of straight lines

Question

If one of the lines $4 x^{2} - cxy + y^{2} = 0$ is $x + y = 0$ then $c =$

Easy

A

$- \frac{1}{5}$
B

$\frac{1}{5}$
C

5
D

-5

Solution

If $m_{1}, m_{2}$ are slopes of lines represented by the combined equation ${ax}^{2} + 2 hxy + {by}^{2} = 0$ then the equation whose roots are $m_{1}, m_{2}$ is ${bm}^{2} + 2 hm + a = 0$
The slope of the line $x + y = 0$ is $- 1$
Hence -1 is the root of $m^{2} - cm + 4 = 0$
It implies that $1 + c + 4 = 0 \Rightarrow c = - 5$

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