Aspire Faculty ID #17913 · Topic: JEE Main 2026 (21 January Evening Shift) · Just now
JEE Main 2026 (21 January Evening Shift)

In the line $ax + 4y = \sqrt{7}$, where $a \in R$, touches the ellipse $3x^2 + 4y^2 = 1$ at the point $P$ in the first quadrant, then one of the focal distances of $P$ is:

Solution

$ax + 4y - \sqrt{7} = 0$ touches $3x^2 + 4y^2 = 1$ 
$\Rightarrow \frac{c^2}{a^2} = \frac{7}{16} = \frac{1}{3}\cdot \frac{a^2}{16} + \frac{1}{4}$ 
$\Rightarrow a = 3, -3$ 
Tangent is $3x + 4y - \sqrt{7} = 0$ 
Let the point of contact is $P(x_1, y_1)$ 
$\Rightarrow 3x_1 + 4y_1 = \sqrt{7}$ 
$\Rightarrow \frac{3x_1}{3} + \frac{4y_1}{4} = \frac{\sqrt{7}}{1}$ 
$\Rightarrow P\left(\frac{1}{\sqrt{7}}, \frac{1}{\sqrt{7}}\right)$ 
$e = \sqrt{1 - \frac{3}{4}} = \frac{1}{2}$ 
$PS = e \cdot PM$ 
$\Rightarrow PS = \frac{1}{\sqrt{3}} + \frac{1}{2\sqrt{7}}$
← Previous Next →

Previous 10 Questions — JEE Main 2026 (21 January Evening Shift)

Nearest first
1
Let $f(x) = x^3 + x^2 f'(1) + 2x f''(2) + f'''(3)$, $x \in R$. Then the value of $f'(5)$ is:
Topic: JEE Main 2026 (21 January Evening Shift)
 2
Let $f : R \to R$ be a twice differentiable function such that $f''(x) > 0$ for all $x \in R$ and $f'(a-1) = 0$, whe…
Topic: JEE Main 2026 (21 January Evening Shift)
 3
The positive integer $n$, for which the solutions of the equation $x(x+2) + (x+2)(x+4) + \cdots + (x+2n-2)(x+2n) = \fra…
Topic: JEE Main 2026 (21 January Evening Shift)

Next 10 Questions — JEE Main 2026 (21 January Evening Shift)

Ascending by ID
1
Let $y^2 = 12x$ be the parabola with its vertex at $O$. Let $P$ be a point on the parabola and $A$ be a point on the y-…
Topic: JEE Main 2026 (21 January Evening Shift)
 2
Let one end of a focal chord of the parabola $y^2 = 16x$ be $(16, 16)$. If $P(\alpha, \beta)$ divides this focal chord …
Topic: JEE Main 2026 (21 January Evening Shift)
 3
Let the line $L$ pass through the point $(-3, 5, 2)$ and make equal angles with the positive coordinate axes. If the di…
Topic: JEE Main 2026 (21 January Evening Shift)
 4
Let the line $L_1$ be parallel to the vector $-3\hat{i} + 2\hat{j} + 4\hat{k}$ and pass through the point $(2, 6, 7)$, …
Topic: JEE Main 2026 (21 January Evening Shift)
 5
Let $\alpha$ and $\beta$ be the roots of equation $x^2 + 2ax + (3a + 10) = 0$ such that $\alpha < 1 < \beta$. Then the …
Topic: JEE Main 2026 (21 January Evening Shift)
 6
A random variable $X$ takes values $0, 1, 2, 3$ with probabilities $\frac{2a+1}{30}, \frac{8a-1}{30}, \frac{4a+1}{30},b…
Topic: JEE Main 2026 (21 January Evening Shift)
 7
If the area of the region ${(x,y) : 1 - 2x \le y \le 4 - x^2,; x \ge 0,; y \ge 0}$ is $\frac{\alpha}{\beta}$, $\alpha, …
Topic: JEE Main 2026 (21 January Evening Shift)
 8
Let $a_1, \frac{a_2}{2}, \frac{a_3}{2^2},; \ldots, \frac{a_{10}}{2^9}$ be a G.P. of common ratio $\frac{1}{\sqrt{2}}$. …
Topic: JEE Main 2026 (21 January Evening Shift)
 9
Let $A = {x : |x^2 - 10| \le 6}$ and $B = {x : |x - 2| > 1}$. Then
Topic: JEE Main 2026 (21 January Evening Shift)
 10
For the matrices $A = \begin{bmatrix} 3 & -4 \ 1 & -1 \end{bmatrix}, \quad B = \begin{bmatrix} -29 & 49 \ -13 & 18 \en…
Topic: JEE Main 2026 (21 January Evening Shift)
Ask Your Question or Put Your Review.

loading...