The equations of the sides AB and AC of a triangle ABC are ( + 1 ) x + y = 4 and x + ( 1 - ) y + = 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 parabola y 2 = 6 x

STUDY MATERIALS
COURSES
MORE
SUCCESS STORY
ADMISSION
STORE

Back

JEE ADVANCE

CLASS XI - XII
CRASH COURSE

CLASS XI - XII

CRASH COURSE

JEE MAIN

CLASS XI - XII
CRASH COURSE

CLASS XI - XII

CRASH COURSE

NEET

CLASS XI - XII
CRASH COURSE

CLASS XI - XII

CRASH COURSE

BITSAT

CLASS XI - XII

CLASS XI - XII

CBSE BOARD

CLASS IX

CLASS X

CLASS XI

CLASS XII

CLASS VI

CLASS VII

CLASS VIII

UP BOARD

CLASS IX
CLASS X
CLASS XI
CLASS XII

CLASS IX

CLASS X

CLASS XI

CLASS XII

CUET

CRASH COURSE

CRASH COURSE

Courses

CLASSROOM COURSES
ONLINE COURSES

ONLINE COURSES

More

Solution

Q.
The equations of the sides AB and AC of a triangle ABC are $(λ + 1) x + λ y = 4 and λ x + (1 - λ) y + λ = 0$ 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 be part of the parabola $y^{2} = 6 x$ in the first quadrant is [2023]

1 $2 \sqrt{2}$

2 $2$

3 $\sqrt{6}$

4 $4$

Ans.
(1)

$A B : (λ + 1) x + λ y = 4$

$A C : λ x + (1 - λ) y + λ = 0$

Since vertex $A$ is on $y$ -axis $\Rightarrow x = 0$

$So, y = \frac{4}{λ} and y = \frac{λ}{λ - 1}$
$\Rightarrow \frac{4}{λ} = \frac{λ}{λ - 1} \Rightarrow λ = 2$

$A B : 3 x + 2 y = 4$

$A C : 2 x - y + 2 = 0 \Rightarrow A (0, 2)$

Let coordinates of $C$ be $(α, 2 α + 2)$ .

$Now, [slope of altitude through C] \times (\frac{- 3}{2}) = - 1$

$\Rightarrow (\frac{2 α}{α - 1}) (\frac{- 3}{2}) = - 1 \Rightarrow α = \frac{- 1}{2}$

So, coordinates of $C$ are $(\frac{- 1}{2}, 1) .$

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

$∴$ $m = 1, - 3$

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

$T \equiv (\frac{a}{m^{2}}, \frac{2 a}{m}) \equiv (\frac{3}{2}, 3)$

$∴$ $C T = \sqrt{{(\frac{3}{2} + \frac{1}{2})}^{2} + {(3 - 1)}^{2}} = \sqrt{4 + 4} = 2 \sqrt{2}$

Want Us to Email you About Special Offers & Updates?