TS EAMCET · Maths · Straight Lines

A quadrilateral $A B C D$ is divided by the diagonal $A C$ into two triangles of equal areas. If $A, B, C$ are respectively $(3, 4), (- 3, 6), (- 5, 1)$ , then the locus of $D$ is

A $(x - 8 y - 57) (x - 8 y + 11) = 0$
B $(x - 8 y - 57) (x - 8 y - 11) = 0$
C $(3 x - 8 y - 57) (3 x - 8 y + 11) = 0$
D $(3 x - 8 y - 11) (3 x - 8 y + 57) = 0$

Verified Solution

Answer & Solution

Correct Answer

(D) $(3 x - 8 y - 11) (3 x - 8 y + 57) = 0$

Step-by-step Solution

Detailed explanation

Given points are A(3,4), B(-3,6), C(-5,1), D(x,y). Here area of ABC=Area of ACD. Area of ABC=12∑x1y2-y3 =1236-1+-31-4+-54-6 =1235+-3-3+-5-2 =1215+9+10 =17 ...1 Area of ACD=12∑x1y2-y3 =1231-y+-5y-4+x4-1 =123-3y-5y+20+3x…

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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A quadrilateral ABCD is divided by the diagonal AC into two triangles of equal areas. If A, B, C are respectively 3,4, -3,6, -5,1, then the locus of D is

Step-by-step Solution

A quadrilateral $A B C D$ is divided by the diagonal $A C$ into two triangles of equal areas. If $A, B, C$ are respectively $(3, 4), (- 3, 6), (- 5, 1)$ , then the locus of $D$ is