The equations of the sides AB and AC of a triangle ABC are ( +1) … - Vidyadip

MCQ

The equations of the sides $AB$ and $AC$ of a triangle $ABC$ are $(\lambda+1) x +\lambda y =4 \text { and } \lambda x +(1-\lambda) y +\lambda=0$ respectively. Its vertex $A$ is on the $y$-axis and its orthocentre is $(1,2)$. The length of the tangent from the point $C$ to the part of the parabola $y^2=6 x$ in the first quadrant is

A
$\sqrt{6}$
✓
$2 \sqrt{2}$
C
$2$
D
$4$

✓

Answer

Correct option: B.

$2 \sqrt{2}$

b
$AB :(\lambda+1) x +\lambda y =4$

$AC : \lambda x +(1-\lambda) y +\lambda=0$

Vertex $A$ is on $y$-axis

$\Rightarrow x=0$

So $y =\frac{4}{\lambda}, y =\frac{\lambda}{\lambda-1}$

$\Rightarrow \frac{4}{\lambda}=\frac{\lambda}{\lambda-1}$

$\Rightarrow \lambda=2$

$AB : 3 x +2 y =4$

$AC : 2 x - y +2=0$

$\Rightarrow A (0,2) \text { Let } C (\alpha, 2 \alpha+2)$

Now (Slope of Altitude through C) $\left(-\frac{3}{2}\right)=-1$

$\left(\frac{2 \alpha}{\alpha-1}\right)\left(-\frac{3}{2}\right)=-1 \Rightarrow \alpha=-\frac{1}{2}$

So $C \left(-\frac{1}{2}, 1\right)$

Let Equation of tangent be $y=m x+\frac{3}{2 m}$

$m ^2+2 m -3=0$

$\Rightarrow m =1,-3$

So tangent which touches in first quadrant at $T$ is

$T \equiv\left(\frac{ a }{ m ^2}, \frac{2 a }{ m }\right)$

$\equiv\left(\frac{3}{2}, 3\right)$

$\Rightarrow CT =\sqrt{4+4}=2 \sqrt{2}$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "MCQ" from 10.2 parabola,ellipse,hyperbola→10.2 parabola,ellipse,hyperbola — all question types→MATHS STD 11 Science question papers→Gujarat Board STD 11 Science question papers→Gujarat Board question paper generator→

Similar questions

What is the sum of 1 + 2 + 3 + ... n?

Let $\Gamma$ be a circle with diameter $A B$ and centre $O$. Let $l$ be the tangent to $\Gamma$ at $B$. For each point $M$ on $\Gamma$ different from $A$, consider the tangent $t$ at $M$ and let interest $l$ at $P$. Draw a line parallel to $A B$ through $P$ intersecting $M M$ at $Q$. The locus of $Q$ as $M$ varies over $\Gamma$ is

In a town of $10,000$ families it was found that $40\%$ family buy newspaper $A, 20\%$ buy newspaper $B$ and $10\%$ families buy newspaper $C, 5\%$ families buy $A$ and $B, 3\%$ buy $B$ and $C$ and $4\%$ buy $A$ and $C$. If $2\%$ families buy all the three newspapers, then number of families which buy $A$ only is

The conditions that $ax + by + c = 0$ is tangent to the parabola $y^2 = 4ax$, is

Two parabolas $y^2 = 4a(x - l_1) $ and $x^2 = 4a (y - l_2) $ always touch one another, the quantities $ l_1$ and $l_2 $ are both variable. Locus of their point of contact has the equation

Let $z \in C$ the set of complex numbers. Then the equation, $2\left| {z + 3i} \right| - \left| {z - i} \right| = 0$ represents

The locus of the point of instruction of the lines $\sqrt 3 x - y - 4 \sqrt 3 t = 0$ $\&$ $\sqrt 3tx + ty - 4\sqrt 3 = 0$ (where $ t$ is a parameter) is a hyperbola whose eccentricity is

An ellipse passes through the foci of the hyperbola, $9x^2 - 4y^2 = 36$ and its major and minor axes lie along the transverse and conjugate axes of the hyperbola respectively. If the product of eccentricities of the two conics is $\frac {1}{2}$, then which of the following points does not lie on the ellipse?

If A (6, 4) and B (2, 12) are the two points, then the slope of a line perpendicular to line AB is:

Let the mean and standard deviation of marks of class $A$ of $100$ students be respectively $40$ and $\alpha( > 0)$, and the mean and standard deviation of marks of class $B$ of $n$ students be respectively $55$ and $30-\alpha$. If the mean and variance of the marks of the combined class of $100+ n$ students are respectively $50$ and $350$,then the sum of variances of classes $A$ and $B$ is