Question

$\text{Assertion}$

$A$: lf two sides of a triangle are represented by $2x^2+4xy-y^2=0$ and orthocentre is $(1,1)$, then the third side is $x+y+3=0$.

$\text{Reason}$

$R$: lf two sides of a triangle are represented by $a{x}^2+2hxy+b{y}^2=0$ and orthocentre is $(c,d)$, then the third side is $(a+b)(cx+dy)=a{d}^2-2hcd+b{c}^2$.

• A. Both A and R are true and R is the correct explanation of A.
• B. Both A and R are true and R is not correct explanation of A.
• C. A is true but R is false.
• D. A is false but R is true.

Answer 1

Answer: (A)

Both A and R are true and R is the correct explanation of A.

Assertion: $2x^2+4xy-y^2=0$

$t=\frac{y}{x}$

$-t^2+4t+2=0$

$t^2-4t-2=0$

$t=\frac{4\pm \sqrt{24}}{2}$

Using the reason statement, third side is

$(2-1)(x-y)=2-2\times 2+(-1)1$

$x+y=-3$

$x+y+3=0$

Hence, option A, as Reason is the correct explanation of Assertion.

$\text{Point to remember}$: If two sides of a triangle represented by $a{x}^2+2hxy+b{y}^2=0$ and orthocentre is $(c,d)$ then the third side is $(a+b)(cx+dy)=a{d}^2-2hcd+b{c}^2$