TS EAMCET · Maths · Parabola

If $x - 2 y + k = 0$ is a tangent to the parabola $y^{2} - 4 x - 4 y + 8 = 0,$ then the slope of the tangent drawn at $(1, k)$ on the given parabola is

A $\frac{- 5}{2}$
B $2$
C $- 2$
D $\frac{2}{5}$

Verified Solution

Answer & Solution

Correct Answer

(D) $\frac{2}{5}$

Step-by-step Solution

Detailed explanation

Given: x-2y+k=0 is a tangent to the parabola y2-4x-4y+8=0 Parabola can be written y-22=4x-1 Equation of tangent in slope form y-2=mx-1+am ⇒y=mx-m+1m+2 ...1 a=1 Now given tangent y=x2+k2 ...2 Using Equations 1 & 2…

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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If x-2y+k=0 is a tangent to the parabola y2-4x-4y+8=0, then the slope of the tangent drawn at 1, k on the given parabola is

Step-by-step Solution

If $x - 2 y + k = 0$ is a tangent to the parabola $y^{2} - 4 x - 4 y + 8 = 0,$ then the slope of the tangent drawn at $(1, k)$ on the given parabola is