TS EAMCET · Maths · Straight Lines

If the two lines $\frac{3}{2} x + (2 a - 1) y = 3$ and $2 x + a^{2} y = - 3$ are perpendicular, then the distance of their point of intersection from the point $(1, 1)$ is

A $\frac{9}{\sqrt{5}}$
B $3$
C $\frac{\sqrt{45}}{5}$
D $\sqrt{\frac{3}{5}}$

Verified Solution

Answer & Solution

Correct Answer

(B) $3$

Step-by-step Solution

Detailed explanation

Given that, Two lines L1=3x2+2a-1y=3 and L2=2x+a2y=-3 are perpendicular. L1⊥L2 so product of their gradients will be -1 Let gradients of L1=m1and L2=m2 Gradient of L1=m1=-322a-1 Gradient of L2=m2=-2a2 L1⊥L2 ⇒m1×m2=-1 …

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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Step-by-step Solution

If the two lines $\frac{3}{2} x + (2 a - 1) y = 3$ and $2 x + a^{2} y = - 3$ are perpendicular, then the distance of their point of intersection from the point $(1, 1)$ is