First slide

Pair of straight lines

Question

The area of the triangle formed by the lines $y = x + k, {ax}^{2} + 2 hxy + {by}^{2} = 0$ is

Moderate

A

$\frac{k^{2} \sqrt{h^{2} - ab}}{|b + 2 h - a|}$
B

$\frac{k^{2} \sqrt{h^{2} - ab}}{|a + 2 h - b|}$
C

$\frac{k^{2} \sqrt{h^{2} - ab}}{|a + 2 h + b|}$
D

$\frac{k^{2} \sqrt{h^{2} - ab}}{|a - 2 h + b|}$

Solution

The area of the triangle formed by the lines ${ax}^{2} + 2 hxy + {by}^{2} = 0$ and $lx + my + n = 0$ is $|\frac{n^{2} \sqrt{h^{2} - ab}}{{am}^{2} - 2 hlm + {bl}^{2}}|$
Since the equation is $y = x + k, l = 1, m = - 1, n = k$
Therefore, the area of the triangle is $\frac{k^{2} \sqrt{h^{2} - ab}}{|a {(- 1)}^{2} - 2 h (1) (- 1) + b {(1)}^{2}|} = |\frac{k^{2} \sqrt{h^{2} - ab}}{a + 2 h + b}|$

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