When the coordinate axes are rotated about the origin in the positive direction through an angle π4, if the equation 49x2+25y2=1225 is transformed to px2+qxy+ry2=t and the G.C.D of p,q,r,t is 1, then

TS EAMCET · Maths · Straight Lines

When the coordinate axes are rotated about the origin in the positive direction through an angle $\frac{π}{4}$ , if the equation $49 x^{2} + 25 y^{2} = 1225$ is transformed to $p x^{2} + q x y + r y^{2} = t$ and the G.C.D of $p, q, r, t$ is $1$ , then

A ${(p - q + r - 32)}^{2} = 4 t$
B ${(p - q - r + 12)}^{2} = t$
C ${(p + q + r - 15)}^{2} = t$
D ${(- p - q + r + 13)}^{2} = t$

Verified Solution

Answer & Solution

Correct Answer

(C) ${(p + q + r - 15)}^{2} = t$

Step-by-step Solution

Detailed explanation

We know that if coordinate axis is rotated then, x=Xcosθ-Ysinθ & y=Xsinθ+Ycosθ Here angle θ=45°, so on putting the value of angle we get, x=X-Y2 & y=X+Y2, Now putting the value in 49x2+25y2=1225 we get,…

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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