JEE Mathematics 2026 Question with Solution

Given:

• In (S1), $A(5,-1)$ and $B(-2,3)$ are two vertices of a triangle and the orthocentre is $H(0,0)$.
• In (S2), $2a, b, c$ are three consecutive terms of an A.P. and the family of lines is $ax+by+c=0$.

Find: Whether statements (S1) and (S2) are correct.

For (S1), verify whether the third vertex $(-4,-7)$ gives orthocentre at the origin.

Slope of $AB$ is

So the altitude from $C$ must have slope

Now slope of the line joining $C(-4,-7)$ to $H(0,0)$ is

Hence, $CH$ is perpendicular to $AB$.

Next, slope of $BC$ is

So the altitude from $A$ must have slope

The slope of $AH$ is

Hence, $AH$ is perpendicular to $BC$.

Also, slope of $CA$ is

So the altitude from $B$ must have slope

The slope of $BH$ is

Hence, $BH$ is perpendicular to $CA$.

Therefore, all three altitudes pass through $H(0,0)$, so (S1) is correct.

For (S2), since $2a, b, c$ are consecutive terms of an A.P.,

Therefore,

Now test the point $(2,-2)$ in the line

Substituting $x=2$ and $y=-2$,

Using $2b=2a+c$,

So the equation is satisfied identically. Hence every such line passes through $(2,-2)$, so (S2) is correct.

Therefore, both statements are correct. The correct option is A.