If 2x-y+1=0 is a tangent to the hyperbola x2a2-y216=1 , then which of the following CANNOT be sides of a right angled triangle?

JEE Advanced · Mathematics · 15. Hyperbola

If $2 x - y + 1 = 0$ is a tangent to the hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{16} = 1$ , then which of the following CANNOT be sides of a right angled triangle?

A $a, 4, 2$
B $2 a, 8, 1$
C $2 a, 4, 1$
D $a, 4, 1$

Verified Solution

Answer & Solution

Correct Answer

(A) $a, 4, 2$

Step-by-step Solution

Detailed explanation

The line

y = m x + c

is tangent to hyperbola

\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1,

, if

c^{2} = a^{2} m^{2} - b^{2}

∴ {(1)}^{2} = 4 a^{2} - 16 \Rightarrow a^{2} = \frac{17}{4}

\Rightarrow a = \frac{\sqrt{17}}{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.

Same subject

Let $I=\left(\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right)$ and $P=\left(\begin{array}{ll}2 & 0 \\ 0 & 3\end{array}\right)$. Let $Q=\left(\begin{array}{ll}\mathrm{x} & \mathrm{y} \\ \mathrm{z} & 4\end{array}\right)$ for some non-zero real numbers $x, y$, and $z$, for which there is $2 \times 2$ matrix $R$ with all entries being non-zero real numbers, such that $Q R=R P$. Then which of the following statements is (are) TRUE?

JEE Advanced 2025 Medium

Let

ℓ_{1}

and

ℓ_{2}

be the lines

{\vec{r}}_{1} = λ (\hat{i} + \hat{j} + \hat{k})

and

{\vec{r}}_{2} = (\hat{j} - \hat{k}) + μ (\hat{i} + \hat{k})

, respectively, Let

X

be the set of all the planes

H

that contain the line

ℓ_{1}

. For a plane

H

, let

d (H)

denote the smallest possible distance between the points of

ℓ_{2}

and

H

. Let

H_{0}

be a plane in

X

for which

d (H_{0})

is the maximum value of

d (H)

H

varies over all planes in

X

. Match each entry in List-I to the correct entries in List-II.

	List-I		List-II
$(P)$	The value of $d (H_{0})$ is	$(1)$	$\sqrt{3}$
$(Q)$	The distance of the point $(0, 1, 2)$ from $H_{0}$ is	$(2)$	$\frac{1}{\sqrt{3}}$
$(R)$	The distance of origin from $H_{0}$ is	$(3)$	$0$
$(S)$	The distance of origin from the point of intersection of planes $y = z, x = 1$ and $H_{0}$ is	$(4)$	$\sqrt{2}$
		$(5)$	$\frac{1}{\sqrt{2}}$

The correct option is

JEE Advanced 2023 Medium

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