Straight Lines
Straight Lines
nta_abhyas_2025
Grade 11
Question:
Let the lengths of the altitudes from the vertices $A(-1,1)$, $B(5,2)$, $C(3,-1)$ of $\triangle ABC$ are $p_1$, $p_2$, $p_3$ units respectively, then the value of $\frac{(z_1)^{-1} + (z_2)^{-1}}{(z_3)^{-1}}$ is equal to
Step-by-Step Solution
Key Concept: Application of area ratio formula for triangles using coordinate geometry and algebraic manipulation
Using the formula for the ratio of areas and applying the given ratio of sides: $\frac{(\frac{x}{2})^2 + (\frac{y}{2})^2}{(\frac{x}{2})^2 \cdot (\frac{y}{2})^2} = \frac{(\frac{\Delta_1}{\Delta_2})^2}{(\frac{\Delta_1}{\Delta_2})^2} = \frac{(\frac{180+1}{160+1})^2}{\frac{180+1}{160+1}} = \frac{\frac{181^2}{161^2}}{\frac{181}{161}} = \frac{181}{161} = \frac{20}{2.5} = 2.5$.
Correct Answer: 2